Transmission apparatus and associated method of encoded data

ABSTRACT

A transmission apparatus includes an encoder that codes a data sequence with a parity check matrix, wherein the data sequence includes a final information bit sequence and virtual information bits, and outputs the final information bit sequence and a parity sequence, as LDPC codes, and a transmitter that transmits the LDPC codes as a transmission data. A column length of the parity check matrix is longer than a total length of the final information bit sequence and the parity sequence, by a length of the virtual information bits that are set to “0” and are not transmitted. The total length of the final information bit sequence and the parity sequence has a sequence length corresponding to a length from a first column to a determined column of the parity check matrix. The encoder generates the LDPC codes by using the first column to the determined column among one or more column(s) of the parity check matrix.

TECHNICAL FIELD

The present invention relates to an encoder, decoder and encoding methodusing a low-density parity-check convolutional code (LDPC-CC) supportinga plurality of coding rates.

BACKGROUND ART

In recent years, attention has been attracted to a low-densityparity-check (LDPC) code as an error correction code that provides higherror correction capability with a feasible circuit scale. Because ofits high error correction capability and ease of implementation, an LDPCcode has been adopted in an error correction coding scheme forIEEE802.11n high-speed wireless LAN systems, digital broadcastingsystems, and so forth.

An LDPC code is an error correction code defined by low-density paritycheck matrix H. An LDPC code is a block code having a block length equalto number of columns N of parity check matrix H (e.g. see Non-PatentLiterature 1, Non-Patent Literature 4, or Non-Patent Literature 11). Arandom-like LDPC code, array LDPC code, and QC-LDPC code (QC:Quasi-Cyclic) are proposed in Non-Patent Literature 2, Non-PatentLiterature 3, and Non-Patent Literature 12, for example.

However, a characteristic of many current communication systems is thattransmission information is collectively transmitted per variable-lengthpacket or frame, as in the case of Ethernet (registered trademark). Aproblem with applying an LDPC code, which is a block code, to a systemof this kind is, for example, how to make a fixed-length LDPC code blockcorrespond to a variable-length Ethernet (registered trademark) frame.With IEEE802.11n, the length of a transmission information sequence andan LDPC code block length are adjusted by executing padding processingor puncturing processing on a transmission information sequence, but itis difficult to avoid a change in the coding rate and redundant sequencetransmission due to padding or puncturing.

In contrast to this kind of LDPC code of block code (hereinafterreferred to as “LDPC-BC: Low-Density Parity-Check Block Code”), LDPC-CC(Low-Density Parity-Check Convolutional Code) allowing encoding anddecoding of information sequences of arbitrary length have beeninvestigated (see Non-Patent Literature 7, for example).

An LDPC-CC is a convolutional code defined by a low-density parity-checkmatrix, and, as an example, parity check matrix H^(T)[0,n] of an LDPC-CCin a coding rate of R=1/2 (=b/c) is shown in FIG. 1. Here, element h₁^((m))(t) of H^(T)[0,n] has a value of 0 or 1. All elements other thanh₁ ^((m))(t) are 0. M represents the LDPC-CC memory length, and nrepresents the length of an LDPC-CC codeword. As shown in FIG. 1, acharacteristic of an LDPC-CC parity check matrix is that it is aparallelogram-shaped matrix in which 1 is placed only in diagonal termsof the matrix and neighboring elements, and the bottom-left andtop-right elements of the matrix are zero.

An LDPC-CC encoder defined by parity check matrix H^(T)[0,n] when h₁⁽⁰⁾(t)=1 and h₂ ⁽⁰⁾ (t)=1 here is represented by FIG. 2. As shown inFIG. 2, an LDPC-CC encoder is composed of M+1 shift registers ofbit-length c and a modulo 2 adder (exclusive OR calculator).Consequently, a characteristic of an LDPC-CC encoder is that it can beimplemented with extremely simple circuitry in comparison with a circuitthat performs generator matrix multiplication or an LDPC-BC encoder thatperforms computation based on backward (forward) substitution. Also,since the encoder in FIG. 2 is a convolutional code encoder, it is notnecessary to divide an information sequence into fixed-length blockswhen encoding, and an information sequence of any length can be encoded.

CITATION LIST Non-Patent Literature

[NPL 1]

R. G. Gallager, “Low-density parity check codes,” IRE Trans. Inform.Theory, IT-8, pp-21-28, 1962

[NPL 2]

D. J. C. Mackay, “Good error-correcting codes based on very sparsematrices,” IEEE Trans. Inform. Theory, vol. 45, no. 2, pp 399-431, March1999

[NPL 3]

J. L. Fan, “Array codes as low-density parity-check codes,” proc. of 2ndInt. Symp. on Turbo Codes, pp. 543-546, September 2000

[NPL 4]

R. D. Gallager, “Low-Density Parity-Check Codes,” Cambridge, Mass.: MITPress, 1963

[NPL 5]

M. P. C. Fossorier, M. Mihaljevic, and H. Imai, “Reduced complexityiterative decoding of low density parity check codes based on beliefpropagation,” IEEE Trans. Commun., vol. 47., no. 5, pp. 673-680, May1999

[NPL 6]

J. Chen, A. Dholakia, E. Eleftheriou, M. P. C. Fossorier, and X.-Yu Hu,“Reduced-complexity decoding of LDPC codes,” IEEE Trans. Commun., vol.53., no. 8, pp. 1288-1299, August 2005

[NPL 7]

A. J. Feltstrom, and K. S. Zigangirov, “Time-varying periodicconvolutional codes with low-density parity-check matrix,” IEEE Trans.Inform. Theory, vol. 45, no. 6, pp. 2181-2191, September 1999

[NPL 8]

IEEE Standard for Local and Metropolitan Area Networks, IEEEP802.16e/D12, October 2005

[NPL 9] J. Zhang, and M. P. C. Fossorier, “Shuffled iterative decoding,”IEEE Trans. Commun., vol. 53, no. 2, pp. 209-213, February 2005

[NPL 10]

S. Lin, D. J. Jr., Costello, “Error control coding: Fundamentals andapplications,” Prentice-Hall

[NPL 11]

Tadashi Wadayama, “Low-Density Parity-Check Code and the decodingmethod”, Triceps

[NPL 12]

M. P. C. Fossorier, “Quasi-cyclic low-density parity-check codes fromcirculant permutation matrices,” IEEE Trans. Inform. Theory, vol. 50,no. 8, pp. 1788-1793, November 2001.

SUMMARY OF INVENTION Technical Problem

However, an LDPC-CC, LDPC-CC encoder and LDPC-CC decoder for supportinga plurality of coding rates in a low computational complexity andproviding data of good received quality have not been sufficientlyinvestigated.

For example, Non-Patent Literature 10 discloses using puncturing tosupport a plurality of coding rates. To support a plurality of codingrates using puncturing, first, a basic code (i.e. mother code) isprepared to generate a coding sequence in the mother code and thenselect non-transmission bits (i.e. puncturing bits) from the codingsequence. Further, by changing the number of non-transmission bits, aplurality of coding rates are supported. By this means, it is possibleto support all coding rates by the encoder and decoder (i.e. mother codeencoder and decoder), so that it is possible to provide an advantage ofreducing the computational complexity (i.e. circuit scale).

In contrast, as a method of supporting a plurality of coding rates,there is a method of providing different codes (i.e. distributed codes)every coding rate. Especially, as disclosed in Non-Patent Literature 8,an LDPC code has a flexibility of being able to provide various codelengths and coding rates easily, and therefore it is a general method tosupport a plurality of coding rates by a plurality of codes. In thiscase, although a use of a plurality of codes has a disadvantage ofproviding a large computational complexity (i.e. circuit scale),compared to a case where a plurality of coding rates are supported bypuncturing, there is an advantage of providing data of excellentreceived quality.

In view of the above, there are few documents that argue a method ofgenerating an LDPC code that can maintain the received quality of databy preparing a plurality of codes to support a plurality of codingrates, while reducing the computational complexity of the encoder anddecoder. If a method of providing an LDPC code to realize this isestablished, it is possible to improve the received quality of data andreduce the computational complexity at the same time, which has beendifficult to realize.

It is therefore an object of the present invention to provide an LDPC-CCencoding method for improving the received quality of data by realizinga plurality of coding rates using a plurality of codes in an LDPC-CCencoder and decoder, and for realizing the encoder and decoder in a lowcomputational complexity.

Solution to Problem

The encoder of the present invention that creates a low-densityparity-check convolutional code of a time (varying) period of g (where gis a natural number) using a parity check polynomial of equation 44 of acoding rate of (q−1)/q (where q is an integer equal to or greater than3), employs a configuration having: a coding rate setting section thatsets a coding rate of (s−1)/s (s≤q); an r-th computing section thatreceives as input information X_(r,i) (where r=1, 2, . . . , q−1) atpoint in time i and outputs a computation result of A_(Xr,k)(D)X_(i)(D)of equation 44; a parity computing section that receives as input parityP_(i−1) at point in time i−1 and outputs a computation result ofB_(k)(D)P(D) of equation 44; an adding section that acquires anexclusive OR of computation results of the first to (q−1)-th computationsections and the computation result of the parity computing section, asparity P_(i) at point in time i; and an information generating sectionthat sets zero between information X_(s,1) and information X_(q−1,1).

The decoder of the present invention that provides a parity check matrixbased on a parity check polynomial of equation 45 of a coding rate of(q−1)/q (where q is an integer equal to or greater than 3) and decodes alow-density parity-check convolutional code of a time varying period ofg (where g is a natural number) using belief propagation, employs aconfiguration having: a log likelihood ratio setting section that setslog likelihood ratios for information from information X_(s,1) toinformation X_(q−1,i) at point in time i (where i is an integer), to aknown value, according to a set coding rate of (s−1)/s (s≤q); and acomputation processing section that performs row processing computationand column processing computation according to the parity check matrixbased on the parity check polynomial of equation 45, using the loglikelihood ratio.

The encoding method of the present invention for encoding a low-densityparity-check convolutional code of a time varying period of g (where gis a natural number) supporting coding rates of (y−1)/y and (z−1)/z(y<z), includes: generating a low-density parity-check convolutionalcode of the coding rate of (z−1)/z using a parity check polynomial ofequation 46; and generating a low-density parity-check convolutionalcode of the coding rate of (y−1)/y using a parity check polynomial ofequation 47.

Advantageous Effect of Invention

According to the encoder and decoder of the present invention, with anLDPC-CC encoder and decoder, it is possible to realize a plurality ofcoding rates in a low computational complexity and provide data of highreceived quality.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 shows an LDPC-CC parity check matrix;

FIG. 2 shows a configuration of an LDPC-CC encoder;

FIG. 3 shows an example of the configuration of an LDPC-CC parity checkmatrix of a time varying period of 4;

FIG. 4A shows parity check polynomials of an LDPC-CC of a time varyingperiod of 3 and the configuration of parity check matrix H of thisLDPC-CC;

FIG. 4B shows the belief propagation relationship of terms relating toX(D) of “check equation #1” to “check equation #3” in FIG. 4A;

FIG. 4C shows the belief propagation relationship of terms relating toX(D) of “check equation #1” to “check equation #6”;

FIG. 5 shows a parity check matrix of a (7, 5) convolutional code;

FIG. 6 shows an example of the configuration of parity check matrix Habout an LDPC-CC of a coding rate of 2/3 and a time varying period of 2;

FIG. 7 shows an example of the configuration of an LDPC-CC parity checkmatrix of a coding rate of 2/3 and a time varying period of m;

FIG. 8 shows an example of the configuration of an LDPC-CC parity checkmatrix of a coding rate of (n−1)/n and a time varying period of m;

FIG. 9 shows an example of the configuration of an LDPC-CC encodingsection;

FIG. 10 is a drawing for explaining a method ofinformation-zero-termination;

FIG. 11 is a block diagram showing the main configuration of an encoderaccording to Embodiment 2 of the present invention;

FIG. 12 is a block diagram showing the main configuration of a firstinformation computing section according to Embodiment 2;

FIG. 13 is a block diagram showing the main configuration of a paritycomputing section according to Embodiment 2;

FIG. 14 is a block diagram showing another main configuration of anencoder according to Embodiment 2;

FIG. 15 is a block diagram showing the main configuration of a decoderaccording to Embodiment 2;

FIG. 16 illustrates operations of a log likelihood ratio setting sectionin a case of a coding rate of 1/2;

FIG. 17 illustrates operations of a log likelihood ratio setting sectionin a case of a coding rate of 2/3;

FIG. 18 shows an example of the configuration of a transmittingapparatus having an encoder according to Embodiment 2;

FIG. 19 shows an example of a transmission format;

FIG. 20 shows an example of the configuration of a receiving apparatushaving a decoder according to Embodiment 2;

FIG. 21 shows an example of the frame configuration of a modulationsignal transmitted by communication apparatus #1 that performs hybridARQ according to Embodiment 3 of the present invention;

FIG. 22 shows an example of the frame configuration of a modulationsignal transmitted by communication apparatus #2, which is thecommunicating party of communication apparatus #1, according toEmbodiment 3;

FIG. 23 shows an example of the flow of frames transmitted betweencommunication apparatus #1 and communication apparatus #2, according toEmbodiment 3;

FIG. 24 illustrates data transmitted in frame #2 and frame #2′;

FIG. 25 illustrates a decoding method upon a retransmission;

FIG. 26 shows another example of the flow of frames transmitted betweencommunication apparatus #1 and communication apparatus #2, according toEmbodiment 3;

FIG. 27 illustrates data transmitted in frame #2 and frame #2′;

FIG. 28 illustrates a decoding method upon the first retransmission;

FIG. 29 illustrates data transmitted in frame #2″;

FIG. 30 illustrates a decoding method upon a second retransmission;

FIG. 31 is a block diagram showing the main configuration ofcommunication apparatus #1 according to Embodiment 3; and

FIG. 32 is a block diagram showing the main configuration ofcommunication apparatus #2 according to Embodiment 3.

DESCRIPTION OF EMBODIMENTS

Now, embodiments of the present invention will be described in detailwith reference to the accompanying drawings.

First, before explaining the specific configurations and operations inembodiments, an LDPC-CC of good characteristics will be explained.

(LDPC-CC of good characteristics)

An LDPC-CC of a time varying period of g with good characteristics isdescribed below.

First, an LDPC-CC of a time varying period of 4 with goodcharacteristics will be described. A case in which the coding rate is1/2 is described below as an example.

Consider equations 1-1 to 1-4 as parity check polynomials of an LDPC-CCfor which the time varying period is 4. At this time, X(D) is apolynomial representation of data (information) and P(D) is a paritypolynomial representation. Here, in equations 1-1 to 1-4, parity checkpolynomials have been assumed in which there are four terms in X(D) andP(D) respectively, the reason being that four terms are desirable fromthe standpoint of obtaining good received quality.

[1]

(D ^(a1) +D ^(a2) +D ^(a3) +D ^(a4))X(D)+(D ^(b1) +D ^(b2) +D ^(b3) +D^(b4))P(D)=0  (Equation 1-1)

(D ^(A1) +D ^(A2) +D ^(A3) +D ^(A4))X(D)+(D ^(B1) +D ^(B2) +D ^(B3) +D^(B4))P(D)=0  (Equation 1-2)

(D ^(α1) +D ^(α2) +D ^(α3) +D ^(α4))X(D)+(D ^(β1) +D ^(β2) +D ^(β3) +D^(β4))P(D)=0  (Equation 1-3)

(D ^(E1) +D ^(E2) +D ^(E3) +D ^(E4))X(D)+(D ^(F1) +D ^(F2) +D ^(F3) +D^(F4))P(D)=0  (Equation 1-4)

In equation 1-1, it is assumed that a1, a2, a3 and a4 are integers(where a1≠a2≠a3≠a4, and a1 to a4 are all mutually different). Use of thenotation “X≠Y≠ . . . ≠Z” is assumed to express the fact that X, Y, Z areall mutually different. Also, it is assumed that b1, b2, b3 and b4 areintegers (where b1≠b2≠b3≠b4). A parity check polynomial of equation 1-1is called “check equation #1,” and a sub-matrix based on the paritycheck polynomial of equation 1-1 is designated first sub-matrix H₁.

In equation 1-2, it is assumed that A1, A2, A3, and A4 are integers(where A1≠A2≠A3≠A4). Also, it is assumed that B1, B2, B3, and B4 areintegers (where B1≠B2≠B3≠B4). A parity check polynomial of equation 1-2is called “check equation #2,” and a sub-matrix based on the paritycheck polynomial of equation 1-2 is designated second sub-matrix H₂.

In equation 1-3, it is assumed that α1, α2, α3, and α4 are integers(where α1≠α2≠α3≠α4). Also, it is assumed that β1, β2, β3, and β4 areintegers (where β1≠β2≠β3≠β4). A parity check polynomial of equation 1-3is called “check equation #3,” and a sub-matrix based on the paritycheck polynomial of equation 1-3 is designated third sub-matrix H₃.

In equation 1-4, it is assumed that E1, E2, E3, and E4 are integers(where E1≠E2≠E3≠E4). Also, it is assumed that F1, F2, F3, and F4 areintegers (where F1≠F2≠F3≠F4). A parity check polynomial of equation 1-4is called “check equation #4,” and a sub-matrix based on the paritycheck polynomial of equation 1-4 is designated fourth sub-matrix H₄.

Next, an LDPC-CC of a time varying period of 4 is considered thatgenerates a parity check matrix such as shown in FIG. 3 from firstsub-matrix H₁, second sub-matrix H₂, third sub-matrix H₃, and fourthsub-matrix H₄.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X(D) and P(D), (a1, a2, a3, a4),(b1, b2, b3, b4), (A1, A2, A3, A4), (B1, B2, B3, B4), (α1, α2, α3, α4),(β1, β2, β3, β4), (E1, E2, E3, E4), (F1, F2, F3, F4), in equations 1-1to 1-4 by 4, provision is made for one each of remainders 0, 1, 2, and 3to be included in four-coefficient sets represented as shown above (forexample, (a1, a2, a3, a4)), and to hold true for all the abovefour-coefficient sets.

For example, if orders (a1, a2, a3, a4) of X(D) of “check equation #1”are set as (a1, a2, a3, a4)=(8, 7, 6, 5), remainders k after dividingorders (a1, a2, a3, a4) by 4 are (0, 3, 2, 1), and one each of 0, 1, 2and 3 are included in the four-coefficient set as remainders k.Similarly, if orders (b1, b2, b3, b4) of P(D) of “check equation #1” areset as (b1, b2, b3, b4)=(4, 3, 2, 1), remainders k after dividing orders(b1, b2, b3, b4) by 4 are (0, 3, 2, 1), and one each of 0, 1, 2 and 3are included in the four-coefficient set as remainders k. It is assumedthat the above condition about “remainder” also holds true for thefour-coefficient sets of X(D) and P(D) of the other parity checkequations (“check equation #2,” “check equation #3” and “check equation#4”).

By this means, the column weight of parity check matrix H configuredfrom equations 1-1 to 1-4 becomes 4 for all columns, which enables aregular LDPC code to be formed. Here, a regular LDPC code is an LDPCcode that is defined by a parity check matrix for which each columnweight is equally fixed, and is characterized by the fact that itscharacteristics are stable and an error floor is unlikely to occur. Inparticular, since the characteristics are good when the column weight is4, an LDPC-CC offering good reception performance can be obtained bygenerating an LDPC-CC as described above.

Table 1 shows examples of LDPC-CCs (LDPC-CCs #1 to #3) of a time varyingperiod of 4 and a coding rate of 1/2 for which the above condition about“remainder” holds true. In table 1, LDPC-CCs of a time varying period of4 are defined by four parity check polynomials: “check polynomial #1,”“check polynomial #2,” “check polynomial #3,” and “check polynomial #4.”

TABLE 1 Code Parity check polynomial LDPC-CC Check polynomial #1: #1 ofa time (D⁴⁵⁸ + D⁴³⁵ + D³⁴¹ + 1)X(D) + (D⁵⁹⁸ + D³⁷³ + D⁶⁷ + 1)P(D) = 0varying Check polynomial #2: period of 4 (D²⁸⁷ + D²¹³ + D¹³⁰ + 1)X(D) +(D⁵⁴⁵ + D⁵⁴² + D¹⁰³ + 1)P(D) = 0 and a Check polynomial #3: coding rate(D⁵⁵⁷ + D⁴⁹⁵ + D³²⁶ + 1)X(D) + (D⁵⁶¹ + D⁵⁰² + D³⁵¹ + 1)P(D) = 0 of 1/2Check polynomial #4: (D⁴²⁶ + D³²⁹ + D⁹⁹ + 1)X(D) + (D³²¹ + D⁵⁵ + D⁴² +1)P(D) = 0 LDPC-CC Check polynomial #1: #2 of a time (D⁵⁰³ + D⁴⁵⁴ +D⁴⁹ + 1)X(D) + (D⁵⁶⁹ + D⁴⁶⁷ + D⁴⁰² + 1)P(D) = 0 varying Check polynomial#2: period of 4 (D⁵¹⁸ + D⁴⁷³ + D²⁰³ + 1)X(D) + (D⁵⁹⁸ + D⁴⁹⁹ + D¹⁴⁵ +1)P(D) = 0 and a Check polynomial #3: coding rate (D⁴⁰³ + D³⁹⁷ + D⁶² +1)X(D) + (D²⁹⁴ + D²⁶⁷ + D⁶⁹ + 1)P(D) = 0 of 1/2 Check polynomial #4:(D⁴⁸³ + D³⁸⁵ + D⁹⁴ + 1)X(D) + (D⁴²⁶ + D⁴¹⁵ + D⁴¹³ + 1)P(D) = 0 LDPC-CCCheck polynomial #1: #3 of a time (D⁴⁵⁴ + D⁴⁴⁷ + D¹⁷ + 1)X(D) + (D⁴⁹⁴ +D²³⁷ + D⁷ + 1)P(D) = 0 varying Check polynomial #2: period of 4 (D⁵⁸³ +D⁵⁴⁵ + D⁵⁰⁶ + 1)X(D) + (D³²⁵ + D⁷¹ + D⁶⁶ + 1)P(D) = 0 and a Checkpolynomial #3: coding rate (D⁴³⁰ + D⁴²⁵ + D⁴⁰⁷ + 1)X(D) + (D⁵⁸² + D⁴⁷ +D⁴⁵ + 1)P(D) = 0 of 1/2 Check polynomial #4: (D⁴³⁴ + D³⁵³ + D¹²⁷ +1)X(D) + (D³⁴⁵ + D²⁰⁷ + D³⁸ + 1)P(D) = 0

In the above description, a case in which the coding rate is 1/2 hasbeen described as an example, but a regular LDPC code is also formed andgood received quality can be obtained when the coding rate is (n−1)/n ifthe above condition about “remainder” holds true for four-coefficientsets in information X1(D), X2(D), . . . , Xn−1(D).

In the case of a time varying period of 2, also, it has been confirmedthat a code with good characteristics can be found if the abovecondition about “remainder” is applied. An LDPC-CC of a time varyingperiod of 2 with good characteristics is described below. A case inwhich the coding rate is 1/2 is described below as an example.

Consider equations 2-1 and 2-2 as parity check polynomials of an LDPC-CCfor which the time varying period is 2. At this time, X(D) is apolynomial representation of data (information) and P(D) is a paritypolynomial representation. Here, in equations 2-1 and 2-2, parity checkpolynomials have been assumed in which there are four terms in X(D) andP(D) respectively, the reason being that four terms are desirable fromthe standpoint of obtaining good received quality.

[2]

(D ^(α1) +D ^(α2) +D ^(α3) +D ^(α4))X(D)+(D ^(b1) +D ^(b2) +D ^(b3) +D^(b4))P(D)=0  (Equation 2-1)

(D ^(A1) +D ^(A2) +D ^(A3) +D ^(A4))X(D)+(D ^(B1) +D ^(B2) +D ^(B3) +D^(B4))P(D)=0  (Equation 2-2)

In equation 2-1, it is assumed that a1, a2, a3, and a4 are integers(where a1≠a2≠a3≠a4). Also, it is assumed that b1, b2, b3, and b4 areintegers (where b1≠b2≠b3≠b4). A parity check polynomial of equation 2-1is called “check equation #1,” and a sub-matrix based on the paritycheck polynomial of equation 2-1 is designated first sub-matrix H₁.

In equation 2-2, it is assumed that A1, A2, A3, and A4 are integers(where A1≠A2≠A3≠A4). Also, it is assumed that B1, B2, B3, and B4 areintegers (where B1≠B2≠B3≠B4). A parity check polynomial of equation 2-2is called “check equation #2,” and a sub-matrix based on the paritycheck polynomial of equation 2-2 is designated second sub-matrix H₂.

Next, an LDPC-CC of a time varying period of 2 generated from firstsub-matrix H₁ and second sub-matrix H₂ is considered.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X(D) and P(D), (a1, a2, a3, a4),(b1, b2, b3, b4), (A1, A2, A3, A4), (B1, B2, B3, B4), in equations 2-1and 2-2 by 4, provision is made for one each of remainders 0, 1, 2, and3 to be included in four-coefficient sets represented as shown above(for example, (a1, a2, a3, a4)), and to hold true for all the abovefour-coefficient sets.

For example, if orders (a1, a2, a3, a4) of X(D) of “check equation #1”are set as (a1, a2, a3, a4)=(8, 7, 6, 5), remainders k after dividingorders (a1, a2, a3, a4) by 4 are (0, 3, 2, 1), and one each of 0, 1, 2and 3 are included in the four-coefficient set as remainders k.Similarly, if orders (b1, b2, b3, b4) of P(D) of “check equation #1” areset as (b1, b2, b3, b4)=(4, 3, 2, 1), remainders k after dividing orders(b1, b2, b3, b4) by 4 are (0, 3, 2, 1), and one each of 0, 1, 2 and 3are included in the four-coefficient set as remainders k. It is assumedthat the above condition about “remainder” also holds true for thefour-coefficient sets of X(D) and P(D) of “check equation #2.”

By this means, the column weight of parity check matrix H configuredfrom equations 2-1 and 2-2 becomes 4 for all columns, which enables aregular LDPC code to be formed. Here, a regular LDPC code is an LDPCcode that is defined by a parity check matrix for which each columnweight is equally fixed, and is characterized by the fact that itscharacteristics are stable and an error floor is unlikely to occur. Inparticular, since the characteristics are good when the column weight is8, an LDPC-CC enabling reception performance to be further improved canbe obtained by generating an LDPC-CC as described above.

Table 2 shows examples of LDPC-CCs (LDPC-CCs #1 and #2) of a timevarying period of 2 and a coding rate of 1/2 for which the abovecondition about “remainder” holds true. In table 2, LDPC-CCs of a timevarying period of 2 are defined by two parity check polynomials: “checkpolynomial #1” and “check polynomial #2.”

TABLE 2 Code Parity check polynomial LDPC-CC #1 of a Check polynomial#1: time varying period (D⁵⁵¹ + D⁴⁶⁵ + D⁹⁸ + 1)X(D) + (D⁴⁰⁷ + D³⁸⁶ +D³⁷³ + 1)P(D) = 0 of 2 and a coding rate Check polynomial #2: of 1/2(D⁴⁴³ + D⁴³³ + D⁵⁴ + 1)X(D) + (D⁵⁵⁹ + D⁵⁵⁷ + D⁵⁴⁶ + 1)P(D) = 0 LDPC-CC#2 of a Check polynomial #1: time varying period (D²⁶⁵ + D¹⁹⁰ + D⁹⁹ +1)X(D) + (D²⁹⁵ + D²⁴⁶ + D⁶⁹ + 1)P(D) = 0 of 2 and a coding rate Checkpolynomial #2: of 1/2 (D²⁷⁵ + D²²⁶ + D²¹³ + 1)X(D) + (D²⁹⁸ + D¹⁴⁷ +D⁴⁵ + 1)P(D) = 0

In the above description (LDPC-CCs of a time varying period of 2), acase in which the coding rate is 1/2 has been described as an example,but a regular LDPC code is also formed and good received quality can beobtained when the coding rate is (n−1)/n if the above condition about“remainder” holds true for four-coefficient sets in information X1(D),X2(D), . . . , Xn−1(D).

In the case of a time varying period of 3, also, it has been confirmedthat a code with good characteristics can be found if the followingcondition about “remainder” is applied. An LDPC-CC of a time varyingperiod of 3 with good characteristics is described below. A case inwhich the coding rate is 1/2 is described below as an example.

Consider equations 3-1 to 3-3 as parity check polynomials of an LDPC-CCfor which the time varying period is 3. At this time, X(D) is apolynomial representation of data (information) and P(D) is a paritypolynomial representation. Here, in equations 3-1 to 3-3, parity checkpolynomials are assumed such that there are three terms in X(D) and P(D)respectively.

[3]

(D ^(a1) +D ^(a2) +D ^(a3))X(D)+(D ^(b1) +D ^(b2) +D^(b3))P(D)=0  (Equation 3-1)

(D ^(A1) +D ^(A2) +D ^(A3))X(D)+(D ^(B1) +D ^(B2) +D^(B3))P(D)=0  (Equation 3-2)

(D ^(α1) +D ^(α2) +D ^(α3))X(D)+(D ^(β1) +D ^(β2) +D^(β3))P(D)=0  (Equation 1-3)

In equation 3-1, it is assumed that a1, a2, and a3 are integers (wherea1≠a2≠a3). Also, it is assumed that b1, b2 and b3 are integers (whereb1≠b2≠b3). A parity check polynomial of equation 3-1 is called “checkequation #1,” and a sub-matrix based on the parity check polynomial ofequation 3-1 is designated first sub-matrix H₁.

In equation 3-2, it is assumed that A1, A2 and A3 are integers (whereA1≠A2≠A3). Also, it is assumed that B1, B2 and B3 are integers (whereB1≠B2≠B3). A parity check polynomial of equation 3-2 is called “checkequation #2,” and a sub-matrix based on the parity check polynomial ofequation 3-2 is designated second sub-matrix H₂.

In equation 3-3, it is assumed that a1, a2 and a3 are integers (whereα1≠α2≠α3). Also, it is assumed that β1, β2 and β3 are integers (whereβ1≠β2≠β3). A parity check polynomial of equation 3-3 is called “checkequation #3,” and a sub-matrix based on the parity check polynomial ofequation 3-3 is designated third sub-matrix H₃.

Next, an LDPC-CC of a time varying period of 3 generated from firstsub-matrix H₁, second sub-matrix H₂ and third sub-matrix H₃ isconsidered.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X(D) and P(D), (a1, a2, a3), (b1,b2, b3), (A1, A2, A3), (B1, B2, B3), (α1, α2, α3), (β1, β2, β3), inequations 3-1 to 3-3 by 3, provision is made for one each of remainders0, 1, and 2 to be included in three-coefficient sets represented asshown above (for example, (a1, a2, a3)), and to hold true for all theabove three-coefficient sets.

For example, if orders (a1, a2, a3) of X(D) of “check equation #1” areset as (a1, a2, a3)=(6, 5, 4), remainders k after dividing orders (a1,a2, a3) by 3 are (0, 2, 1), and one each of 0, 1, 2 are included in thethree-coefficient set as remainders k. Similarly, if orders (b1, b2, b3)of P(D) of “check equation #1” are set as (b1, b2, b3)=(3, 2, 1),remainders k after dividing orders (b1, b2, b3) by 3 are (0, 2, 1), andone each of 0, 1, 2 are included in the three-coefficient set asremainders k. It is assumed that the above condition about “remainder”also holds true for the three-coefficient sets of X(D) and P(D) of“check equation #2” and “check equation #3.”

By generating an LDPC-CC as above, it is possible to generate a regularLDPC-CC code in which the row weight is equal in all rows and the columnweight is equal in all columns, without some exceptions. Here,“exceptions” refer to part in the beginning of a parity check matrix andpart in the end of the parity check matrix, where the row weights andcolumns weights are not the same as row weights and column weights ofthe other part. Furthermore, when BP decoding is performed, belief in“check equation #2” and belief in “check equation #3” are propagatedaccurately to “check equation #1,” belief in “check equation #1” andbelief in “check equation #3” are propagated accurately to “checkequation #2,” and belief in “check equation #1” and belief in “checkequation #2” are propagated accurately to “check equation #3.”Consequently, an LDPC-CC with better received quality can be obtained.This is because, when considered in column units, positions at which “1”is present are arranged so as to propagate belief accurately, asdescribed above.

The above belief propagation will be described below using accompanyingdrawings. FIG. 4A shows parity check polynomials of an LDPC-CC of a timevarying period of 3 and the configuration of parity check matrix H ofthis LDPC-CC.

“Check equation #1” illustrates a case in which (a1, a2, a3)=(2, 1, 0)and (b1, b2, b3)=(2, 1, 0) in a parity check polynomial of equation 3-1,and remainders after dividing the coefficients by 3 are as follows:(a1%3, a2%3, a3%3)=(2, 1, 0), (b1%3, b2%3, b3%3)=(2, 1, 0), where “Z %3”represents a remainder after dividing Z by 3.

“Check equation #2” illustrates a case in which (A1, A2, A3)=(5, 1, 0)and (B1, B2, B3)=(5, 1, 0) in a parity check polynomial of equation 3-2,and remainders after dividing the coefficients by 3 are as follows:(A1%3, A2%3, A3%3)=(2, 1, 0), (B1%3, B2%3, B3%3)=(2, 1, 0).

“Check equation #3” illustrates a case in which (α1, α2, α3)=(4, 2, 0)and (β1, β2, β3)=(4, 2, 0) in a parity check polynomial of equation 3-3,and remainders after dividing the coefficients by 3 are as follows:(α1%3, α2%3, α3%3)=(1, 2, 0), (β1%3, β2%3, β3%3)=(1, 2, 0).

Therefore, the example of LDPC-CC of a time varying period of 3 shown inFIG. 4A satisfies the above condition about “remainder”, that is, acondition that (a1%3, a2%3, a3%3), (b1%3, b2%3, b3%3), (A1%3, A2%3,A3%3), (B1%3, B2%3, B3%3), (α1%3, α2%3, α3%3) and ((31%3, (32%3, (33%3)are any of the following: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0),(2, 0, 1), (2, 1, 0).

Returning to FIG. 4A again, belief propagation will now be explained. Bycolumn computation of column 6506 in BP decoding, for “1” of area 6501of “check equation #1,” belief is propagated from “1” of area 6504 of“check equation #2” and from “1” of area 6505 of “check equation #3.” Asdescribed above, “1” of area 6501 of “check equation #1” is acoefficient for which a remainder after division by 3 is 0 (a3%3=0(a3=0) or b3%3=0 (b3=0)). Also, “1” of area 6504 of “check equation #2”is a coefficient for which a remainder after division by 3 is 1 (A2%3=1(A2=1) or B2%3=1 (B2=1)). Furthermore, “1” of area 6505 of “checkequation #3” is a coefficient for which a remainder after division by 3is 2 (α2%3=2 (α2=2) or β2%3=2 (β2=2)).

Thus, for “1” of area 6501 for which a remainder is 0 in thecoefficients of “check equation #1,” in column computation of column6506 in BP decoding, belief is propagated from “1” of area 6504 forwhich a remainder is 1 in the coefficients of “check equation #2” andfrom “1” of area 6505 for which a remainder is 2 in the coefficients of“check equation #3.”

Similarly, for “1” of area 6502 for which a remainder is 1 in thecoefficients of “check equation #1,” in column computation of column6509 in BP decoding, belief is propagated from “1” of area 6507 forwhich a remainder is 2 in the coefficients of “check equation #2” andfrom “1” of area 6508 for which a remainder is 0 in the coefficients of“check equation #3.”

Similarly, for “1” of area 6503 for which a remainder is 2 in thecoefficients of “check equation #1,” in column computation of column6512 in BP decoding, belief is propagated from “1” of area 6510 forwhich a remainder is 0 in the coefficients of “check equation #2” andfrom “1” of area 6511 for which a remainder is 1 in the coefficients of“check equation #3.”

A supplementary explanation of belief propagation will now be givenusing FIG. 4B. FIG. 4B shows the belief propagation relationship ofterms relating to X(D) of “check equation #1” to “check equation #3” inFIG. 4A. “Check equation #1” to “check equation #3” in FIG. 4Aillustrate cases in which (a1, a2, a3)=(2, 1, 0), (A1, A2, A3)=(5, 1,0), and (α1, α2, α3)=(4, 2, 0), in terms relating to X(D) of equations3-1 to 3-3.

In FIG. 4B, terms (a3, A3, α3) inside squares indicate coefficients forwhich a remainder after division by 3 is 0, terms (a2, A2, α2) insidecircles indicate coefficients for which a remainder after division by 3is 1, and terms (a1, A1, α1) inside diamond-shaped boxes indicatecoefficients for which a remainder after division by 3 is 2.

As can be seen from FIG. 4B, for a1 of “check equation #1,” belief ispropagated from A3 of “check equation #2” and from α1 of “check equation#3” for which remainders after division by 3 differ; for a2 of “checkequation #1,” belief is propagated from A1 of “check equation #2” andfrom α3 of “check equation #3” for which remainders after division by 3differ; and, for a3 of “check equation #1,” belief is propagated from A2of “check equation #2” and from α2 of “check equation #3” for whichremainders after division by 3 differ. While FIG. 4B shows the beliefpropagation relationship of terms relating to X(D) of “check equation#1” to “check equation #3,” the same applies to terms relating to P(D).

Thus, for “check equation #1,” belief is propagated from coefficientsfor which remainders after division by 3 are 0, 1, and 2 amongcoefficients of “check equation #2.” That is to say, for “check equation#1,” belief is propagated from coefficients for which remainders afterdivision by 3 are all different among coefficients of “check equation#2.” Therefore, beliefs with low correlation are all propagated to“check equation #1.”

Similarly, for “check equation #2,” belief is propagated fromcoefficients for which remainders after division by 3 are 0, 1, and 2among coefficients of “check equation #1.” That is to say, for “checkequation #2,” belief is propagated from coefficients for whichremainders after division by 3 are all different among coefficients of“check equation #1.” Also, for “check equation #2,” belief is propagatedfrom coefficients for which remainders after division by 3 are 0, 1, and2 among coefficients of “check equation #3.” That is to say, for “checkequation #2,” belief is propagated from coefficients for whichremainders after division by 3 are all different among coefficients of“check equation #3.”

Similarly, for “check equation #3,” belief is propagated fromcoefficients for which remainders after division by 3 are 0, 1, and 2among coefficients of “check equation #1.” That is to say, for “checkequation #3,” belief is propagated from coefficients for whichremainders after division by 3 are all different among coefficients of“check equation #1.” Also, for “check equation #3,” belief is propagatedfrom coefficients for which remainders after division by 3 are 0, 1, and2 among coefficients of “check equation #2.” That is to say, for “checkequation #3,” belief is propagated from coefficients for whichremainders after division by 3 are all different among coefficients of“check equation #2.”

By providing for the orders of parity check polynomials of equations 3-1to 3-3 to satisfy the above condition about “remainder” in this way,belief is necessarily propagated in all column computations, so that itis possible to perform belief propagation efficiently in all checkequations and further increase error correction capability.

A case in which the coding rate is 1/2 has been described above for an

LDPC-CC of a time varying period of 3, but the coding rate is notlimited to 1/2. A regular LDPC code is also formed and good receivedquality can be obtained when the coding rate is (n−1)/n (where n is aninteger equal to or greater than 2) if the above condition about“remainder” holds true for three-coefficient sets in information X1(D),X2(D), . . . , Xn−1(D).

A case in which the coding rate is (n−1)/n (where n is an integer equalto or greater than 2) is described below.

Consider equations 4-1 to 4-3 as parity check polynomials of an LDPC-CCfor which the time varying period is 3. At this time, X1(D), X2(D), . .. , Xn−1(D) are polynomial representations of data (information) X1, X2,. . . , Xn−1, and P(D) is a polynomial representation of parity. Here,in equations 4-1 to 4-3, parity check polynomials are assumed such thatthere are three terms in X1(D), X2(D), . . . , Xn−1(D), and P(D)respectively.

[4]

(D ^(a1,1) +D ^(a1,2) +D ^(a1,3))X ₁(D)+(D ^(a2,1) +D ^(a2,2) +D^(a2,3))X ₂(D)+ . . . +(D ^(an−1,1) +D ^(an−1,2) +D ^(an−1,3))X_(n−1)(D)+(D ^(b1) +D ^(b2) +D ^(b3))P(D)=0  (Equation 4-1)

(D ^(A1,1) +D ^(A1,2) +D ^(A1,3))X ₁(D)+(D ^(A2,1) +D ^(A2,2) +D^(A2,3))X ₂(D)+ . . . +(D ^(An−1,1) +D ^(An−1,2) +D ^(An−1,3))X_(n−1)(D)+(D ^(B1) +D ^(B2) +D ^(B3))P(D)=0  (Equation 4-2)

(D ^(α1,1) +D ^(α1,2) +D ^(α1,3))X ₁(D)+(D ^(α2,1) +D ^(α2,2) +D^(α2,3))X ₂(D)+ . . . +(D ^(αn−1,1) +D ^(αn−1,2) +D ^(αn−1,3))X_(n−1)(D)+(D ^(β1) +D ^(β2) +D ^(β3))P(D)=0  (Equation 4-3)

In equation 4-1, it is assumed that a_(i,1), a_(i,2), and a_(i,3) (wherei=1,2, . . . , n−1) are integers (where a_(i,1)≠a_(i,2)≠a_(i,3)). Also,it is assumed that b1, b2 and b3 are integers (where b1≠b2≠b3). A paritycheck polynomial of equation 4-1 is called “check equation #1,” and asub-matrix based on the parity check polynomial of equation 4-1 isdesignated first sub-matrix H₁.

In equation 4-2, it is assumed that A_(i,1), A_(i,2), and A_(i,3) (wherei=1, 2, . . . , n−1) are integers (where A_(i,1)≠A_(i,2)≠A_(i,3)). Also,it is assumed that B1, B2 and B3 are integers (where B1≠B2≠B3). A paritycheck polynomial of equation 4-2 is called “check equation #2,” and asub-matrix based on the parity check polynomial of equation 4-2 isdesignated second sub-matrix H₂.

In equation 4-3, it is assumed that α_(i,1), α_(i,2), and α_(i,3) (wherei=1, 2, . . . , n−1) are integers (where α_(i,1)≠α_(i,2)≠α_(i,3)). Also,it is assumed that β1, β2 and β3 are integers (where β1≠β2≠β3). A paritycheck polynomial of equation 4-3 is called “check equation #3,” and asub-matrix based on the parity check polynomial of equation 4-3 isdesignated third sub-matrix H₃.

Next, an LDPC-CC of a time varying period of 3 generated from firstsub-matrix H₁, second sub-matrix H₂ and third sub-matrix H₃ isconsidered.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X1(D), X2(D), . . . , Xn−1(D), andP(D), (a_(1,1), a_(1,2), a_(1,3)), (a_(2,1), a_(2,2), a_(2,3)), . . . ,(a_(n−1,1), a_(n−1,2), a_(n−1,3)), (b1, b2, b3), (A_(1,1), A_(1,2),A_(1,3)), (A_(2,1), A_(2,2), A_(2,3)), . . . , (A_(n−1,1), A_(n−1,3)),(B1, B2, B3), (α_(1,1), α_(1,2), α_(1,3)), (α_(2,1), α_(2,2), α_(2,3)),. . . , (a_(n−1,1), a_(n−1,2), a_(n−1,3)), (β1, β2, β3), in equations4-1 to 4-3 by 3, provision is made for one each of remainders 0, 1, and2 to be included in three-coefficient sets represented as shown above(for example, (a_(1,1), a_(1,2), a_(1,3))), and to hold true for all theabove three-coefficient sets.

That is to say, provision is made for (a_(1,1)%3, a_(1,2)%3, a_(1,3)%3),(a_(2,1)%3, a_(2,2)%3, a_(2,3)%3), (a_(n−1,1)%3, a_(n−1,2)%3,a_(n−1,3)%3), (b1%3, b2%3, b3%3), (A_(1,1)%3, A_(1,2)%3, A_(1,3)%3),(A_(2,1)%3, A_(2,2)%3, A_(2,3)%3), . . . , (A_(n−1,1)%3, A_(n−1,2)%3,A_(n−1,3)%3), (B1%3, B2%3, B3%3), (α_(1,1)%3, α_(1,2)%3, α_(1,3)%3),(α_(2,1)%3, α_(2,2)%3, α_(2,3)%3), (α_(n−1,1)%3, α_(n−1,2)%3,α_(n−1,3)%3) and (ƒ1%3, β2%3, β3%3) to be any of the following: (0, 1,2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1), (2, 1, 0).

Generating an LDPC-CC in this way enables a regular LDPC-CC code to begenerated. Furthermore, when BP decoding is performed, belief in “checkequation #2” and belief in “check equation #3” are propagated accuratelyto “check equation #1,” belief in “check equation #1” and belief in“check equation #3” are propagated accurately to “check equation #2,”and belief in “check equation #1” and belief in “check equation #2” arepropagated accurately to “check equation #3.” Consequently, an LDPC-CCwith better received quality can be obtained in the same way as in thecase of a coding rate of 1/2.

Table 3 shows examples of LDPC-CCs (LDPC-CCs #1, #2, #3, #4, and #5) ofa time varying period of 3 and a coding rate of 1/2 for which the above“remainder” related condition holds true. In table 3, LDPC-CCs of a timevarying period of 3 are defined by three parity check polynomials:“check (polynomial) equation #1,” “check (polynomial) equation #2” and“check (polynomial) equation #3.”

TABLE 3 Code Parity check polynomial LDPC-CC #1 Check polynomial #1:(D⁴²⁸ + D³²⁵ + 1)X(D) + (D⁵³⁸ + D³³² + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁵³⁸ + D³⁸⁰ + 1)X(D) + (D⁴⁴⁹ + D¹ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁵⁸³ + D¹⁷⁰ + 1)X(D) + (D³⁶⁴ + D²⁴² +1)P(D) = 0 of 3 and a coding rate of 1/2 LDPC-CC #2 Check polynomial #1:(D⁵⁶² + D⁷¹ + 1)X(D) + (D³²⁵ + D¹⁵⁵ + 1)P(D) = 0 of a time Checkpolynomial #2: D²¹⁵ + D¹⁰⁶ + 1)X(D) + (D⁵⁶⁶ + D¹⁴² + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁵⁹⁰ + D⁵⁵⁹ + 1)X(D) + (D¹²⁷ + D¹¹⁰ +1)P(D) = 0 of 3 and a coding rate of 1/2 LDPC-CC #3 Check polynomial #1:(D¹¹² + D⁵³ + 1)X(D) + (D¹¹⁰ + D⁸⁸ + 1)P(D) = 0 of a time Checkpolynomial #2: (D¹⁰³ + D⁴⁷ + 1)X(D) + (D⁸⁵ + D⁸³ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D¹⁴⁸ + D⁸⁹ + 1)X(D) + (D¹⁴⁶ + D⁴⁹ + 1)P(D)= 0 of 3 and a coding rate of 1/2 LDPC-CC #4 Check polynomial #1:(D³⁵⁰ + D³²² + 1)X(D) + (D⁴⁴⁸ + D³³⁸ + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁵²⁹ + D³² + 1)X(D) + (D²³⁸ + D¹⁸⁸ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁵⁹² + D⁵⁷² + 1)X(D) + (D⁵⁷⁸ + D⁵⁶⁸ +1)P(D) = 0 of 3 and a coding rate of 1/2 LDPC-CC #5 Check polynomial #1:(D⁴¹⁰ + D⁸² + 1)X(D) + (D⁸³⁵ + D⁴⁷ + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁸⁷⁵ + D⁷⁹⁶ + 1)X(D) + (D⁹⁶² + D⁸⁷¹ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁶⁰⁵ + D⁵⁴⁷ + 1)X(D) + (D⁹⁵⁰ + D⁴³⁹ +1)P(D) = 0 of 3 and a coding rate of 1/2 LDPC-CC #6 Check polynomial #1:(D³⁷³ + D⁵⁶ + 1)X(D) + (D⁴⁰⁶ + D²¹⁸ + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁴⁵⁷ + D¹⁹⁷ + 1)X(D) + (D⁴⁹¹ + D²² + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁴⁸⁵ + D⁷⁰ + 1)X(D) + (D²³⁶ + D¹⁸¹ + 1)P(D)= 0 of 3 and a coding rate of 1/2

It has been confirmed that, as in the case of a time varying period of3, a code with good characteristics can be found if the condition about“remainder” below is applied to an LDPC-CC for which the time varyingperiod is a multiple of 3 (for example, 6, 9, 12, . . . ). An LDPC-CC ofa multiple of a time varying period of 3 with good characteristics isdescribed below. The case of an LDPC-CC of a coding rate of 1/2 and atime varying period of 6 is described below as an example.

Consider equations 5-1 to 5-6 as parity check polynomials of an LDPC-CCfor which the time varying period is 6.

[5]

(D ^(a1,1) +D ^(a1,2) +D ^(a1,3))X(D)+(D ^(b1,1) +D ^(b1,2) +D^(b1,3))P(D)=0  (Equation 5-1)

(D ^(a2,1) +D ^(a2,2) +D ^(a2,3))X(D)+(D ^(b2,1) +D ^(b2,2) +D^(b2,3))P(D)=0  (Equation 5-2)

(D ^(a3,1) +D ^(a3,2) +D ^(a3,3))X(D)+(D ^(b3,1) +D ^(b3,2) +D^(b3,3))P(D)=0  (Equation 5-3)

(D ^(a4,1) +D ^(a4,2) +D ^(a4,3))X(D)+(D ^(b4,1) +D ^(b4,2) +D^(b4,3))P(D)=0  (Equation 5-4)

(D ^(a5,1) +D ^(a5,2) +D ^(a5,3))X(D)+(D ^(b5,1) +D ^(b5,2) +D^(b5,3))P(D)=0  (Equation 5-5)

(D ^(a6,1) +D ^(a6,2) +D ^(a6,3))X(D)+(D ^(b6,1) +D ^(b6,2) +D^(b6,3))P(D)=0  (Equation 5-6)

At this time, X(D) is a polynomial representation of data (information)and P(D) is a parity polynomial representation. With an LDPC-CC of atime varying period of 6, if i %6=k (where k=0, 1, 2, 3, 4, 5) isassumed for parity Pi and information Xi at time i, a parity checkpolynomial of equation 5-(k+1) holds true. For example, if i=1, i %6=1(k=1), and therefore equation 6 holds true.

[6]

(D ^(a2,1) +D ^(a2,2) +D ^(a2,3))X ₁+(D ^(b2,1) +D ^(b2,2) +D ^(b2,3))P₁=0  (Equation 6)

Here, in equations 5-1 to 5-6, parity check polynomials are assumed suchthat there are three terms in X(D) and P(D) respectively.

In equation 5-1, it is assumed that a1,1, a1,2, a1,3 are integers (wherea1,1≠a1,2≠a1,3). Also, it is assumed that b1,1, b1,2, and b1,3 areintegers (where b1,1≠b1,2≠b1,3). A parity check polynomial of equation5-1 is called “check equation #1,” and a sub-matrix based on the paritycheck polynomial of equation 5-1 is designated first sub-matrix H₁.

In equation 5-2, it is assumed that a2,1, a2,2, and a2,3 are integers(where a2,1≠a2,2≠a2,3). Also, it is assumed that b2,1, b2,2, b2,3 areintegers (where b2,1≠b2,2≠b2,3). A parity check polynomial of equation5-2 is called “check equation #2,” and a sub-matrix based on the paritycheck polynomial of equation 5-2 is designated second sub-matrix H₂.

In equation 5-3, it is assumed that a3,1, a3,2, and a3,3 are integers(where a3,1≠a3,2≠a3,3). Also, it is assumed that b3,1, b3,2, and b3,3are integers (where b3,1≠b3,2≠b3,3). A parity check polynomial ofequation 5-3 is called “check equation #3,” and a sub-matrix based onthe parity check polynomial of equation 5-3 is designated thirdsub-matrix H₃.

In equation 5-4, it is assumed that a4,1, a4,2, and a4,3 are integers(where a4,1≠a4,2≠a4,3). Also, it is assumed that b4,1, b4,2, and b4,3are integers (where b4,1≠b4,2≠b4,3). A parity check polynomial ofequation 5-4 is called “check equation #4,” and a sub-matrix based onthe parity check polynomial of equation 5-4 is designated fourthsub-matrix H₄.

In equation 5-5, it is assumed that a5,1, a5,2, and a5,3 are integers(where a5,1≠a5,2≠a5,3). Also, it is assumed that b5,1, b5,2, and b5,3are integers (where b5,1≠b5,2≠b5,3). A parity check polynomial ofequation 5-5 is called “check equation #5,” and a sub-matrix based onthe parity check polynomial of equation 5-5 is designated fifthsub-matrix H₅.

In equation 5-6, it is assumed that a6,1, a6,2, and a6,3 are integers(where a6,1≠a6,2≠a6,3). Also, it is assumed that b6,1, b6,2, and b6,3are integers (where b6,1≠b6,2≠b6,3). A parity check polynomial ofequation 5-6 is called “check equation #6,” and a sub-matrix based onthe parity check polynomial of equation 5-6 is designated sixthsub-matrix H₆.

Next, an LDPC-CC of a time varying period of 6 is considered that isgenerated from first sub-matrix H₁, second sub-matrix H₂, thirdsub-matrix H₃, fourth sub-matrix H₄, fifth sub-matrix H₅ and sixthsub-matrix H₆.

At this time, if k is designated as a remainder after dividing thevalues of combinations of orders of X(D) and P(D), (a1,1, a1,2, a1,3),(b1,1, b1,2, b1,3), (a2,1, a2,2, a2,3), (b2,1, b2,2, b2,3), (a3,1, a3,2,a3,3), (b3,1, b3,2, b3,3), (a4,1, a4,2, a4,3), (b4,1, b4,2, b4,3),(a5,1, a5,2, a5,3), (b5,1, b5,2, b5,3), (a6,1, a6,2, a6,3), (b6,1, b6,2,b6,3), in equations 5-1 to 5-6 by 3, provision is made for one each ofremainders 0, 1, and 2 to be included in three-coefficient setsrepresented as shown above (for example, (a1,1, a1,2, a1,3)), and tohold true for all the above three-coefficient sets. That is to say,provision is made for (a1,1%3, a1,2%3, a1,3%3), (b1,1%3, b1,2%3,b1,3%3), (a2,1%3, a2,2%3, a2,3%3), (b2,1%3, b2,2%3, b2,3%3), (a3,1%3,a3,2%3, a3,3%3), (b3,1%3, b3,2%3, b3,3%3), (a4,1%3, a4,2%3, a4,3%3),(b4,1%3, b4,2%3, b4,3%3), (a5,1%3, a5,2%3, a5,3%3), (b5,1%3, b5,2%3,b5,3%3), (a6,1%3, a6,2%3, a6,3%3) and (b6,1%3, b6,2%3, b6,3%3) to be anyof the following: (0, 1, 2), (0, 2, 1), (1, 0, 2), (1, 2, 0), (2, 0, 1),(2, 1, 0).

By generating an LDPC-CC in this way, if an edge is present when aTanner graph is drawn for “check equation #1,” belief in “check equation#2 or check equation #5” and belief in “check equation #3 or checkequation #6” are propagated accurately.

Also, if an edge is present when a Tanner graph is drawn for “checkequation #2,” belief in “check equation #1 or check equation #4” andbelief in “check equation #3 or check equation #6” are propagatedaccurately.

If an edge is present when a Tanner graph is drawn for “check equation#3,” belief in “check equation #1 or check equation #4” and belief in“check equation #2 or check equation #5” are propagated accurately. Ifan edge is present when a Tanner graph is drawn for “check equation #4,”belief in “check equation #2 or check equation #5” and belief in “checkequation #3 or check equation #6” are propagated accurately.

If an edge is present when a Tanner graph is drawn for “check equation#5,” belief in “check equation #1 or check equation #4” and belief in“check equation #3 or check equation #6” are propagated accurately. Ifan edge is present when a Tanner graph is drawn for “check equation #6,”belief in “check equation #1 or check equation #4” and belief in “checkequation #2 or check equation #5” are propagated accurately.

Consequently, an LDPC-CC of a time varying period of 6 can maintainbetter error correction capability in the same way as when the timevarying period is 3.

In this regard, belief propagation will be described using FIG. 4C. FIG.4C shows the belief propagation relationship of terms relating to X(D)of “check equation #1” to “check equation #6.” In FIG. 4C, a squareindicates a coefficient for which a remainder after division by 3 inax,y (where x=1, 2, 3, 4, 5, 6, and y=1, 2, 3) is 0.

A circle indicates a coefficient for which a remainder after division by3 in ax,y (where x=1, 2, 3, 4, 5, 6, and y=1, 2, 3) is 1. Adiamond-shaped box indicates a coefficient for which a remainder afterdivision by 3 in ax,y (where x=1, 2, 3, 4, 5, 6, and y=1, 2, 3) is 2.

As can be seen from FIG. 4C, if an edge is present when a Tanner graphis drawn, for a1,1 of “check equation #1,” belief is propagated from“check equation #2 or #5” and “check equation #3 or #6” for whichremainders after division by 3 differ. Similarly, if an edge is presentwhen a Tanner graph is drawn, for a1,2 of “check equation #1,” belief ispropagated from “check equation #2 or #5” and “check equation #3 or #6”for which remainders after division by 3 differ.

Similarly, if an edge is present when a Tanner graph is drawn, for a1,3of “check equation #1,” belief is propagated from “check equation #2 or#5” and “check equation #3 or #6” for which remainders after division by3 differ. While FIG. 4C shows the belief propagation relationship ofterms relating to X(D) of “check equation #1” to “check equation #6,”the same applies to terms relating to P(D).

Thus, belief is propagated to each node in a Tanner graph of “checkequation #1” from coefficient nodes of other than “check equation #1.”Therefore, beliefs with low correlation are all propagated to “checkequation #1,” enabling an improvement in error correction capability tobe expected.

In FIG. 4C, “check equation #1” has been focused upon, but a Tannergraph can be drawn in a similar way for “check equation #2” to “checkequation #6,” and belief is propagated to each node in a Tanner graph of“check equation #K” from coefficient nodes of other than “check equation#K.” Therefore, beliefs with low correlation are all propagated to“check equation #K” (where K=2, 3, 4, 5, 6), enabling an improvement inerror correction capability to be expected.

By providing for the orders of parity check polynomials of equations 5-1to 5-6 to satisfy the above condition about “remainder” in this way,belief can be propagated efficiently in all check equations, and thepossibility of being able to further improve error correction capabilityis increased.

A case in which the coding rate is 1/2 has been described above for anLDPC-CC of a time varying period of 6, but the coding rate is notlimited to 1/2. The possibility of obtaining good received quality canbe increased when the coding rate is (n−1)/n (where n is an integerequal to or greater than 2) if the above condition about “remainder”holds true for three-coefficient sets in information X1(D), X2(D), . . ., Xn−1(D).

A case in which the coding rate is (n−1)/n (where n is an integer equalto or greater than 2) is described below.

Consider equations 7-1 to 7-6 as parity check polynomials of an LDPC-CCfor which the time varying period is 6.

[7]

(D ^(a#1,1,1) +D ^(a#1,1,2) +D ^(a#1,1,3))X ₁(D)+(D ^(a#1,2,1) +D^(a#1,2,2) +D ^(a#1,2,3))X ₂(D)+ . . . +(D ^(a#1,n−1,1) +D ^(a#1,n−1,2)+D ^(a#1,n−1,3))X _(n−1)(D)+(D ^(b#1,1) +D ^(b#1,2) +D^(b#1,3))P(D)=0  (Equation 7- 1)

(D ^(a#2,1,1) +D ^(a#2,1,2) +D ^(a#2,1,3))X ₁(D)+(D ^(a#2,2,1) +D^(a#2,2,2) +D ^(a#2,2,3))X ₂(D)+ . . . +(D ^(a#2,n−1,1) +D ^(a#2,n−1,2)+D ^(a#2,n−1,3))X _(n−1)(D)+(D ^(b#2,1) +D ^(b#2,2) +D^(b#2,3))P(D)=0  (Equation 7- 2)

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X ₁(D)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X ₂(D)+ . . . +(D ^(a#3,n−1,1) +D ^(a#3,n−1,2)+D ^(a#3,n−1,3))X _(n−1)(D)+(D ^(b#3,1) +D ^(b#3,2) +D^(b#3,3))P(D)=0  (Equation 7- 3)

(D ^(a#4,1,1) +D ^(a#4,1,2) +D ^(a#4,1,3))X ₁(D)+(D ^(a#4,2,1) +D^(a#4,2,2) +D ^(a#4,2,3))X ₂(D)+ . . . +(D ^(a#4,n−1,1) +D ^(a#4,n−1,2)+D ^(a#4,n−1,3))X _(n−1)(D)+(D ^(b#4,1) +D ^(b#4,2) +D^(b#4,3))P(D)=0  (Equation 7- 4)

(D ^(a#5,1,1) +D ^(a#5,1,2) +D ^(a#5,1,3))X ₁(D)+(D ^(a#5,2,1) +D^(a#5,2,2) +D ^(a#5,2,3))X ₂(D)+ . . . +(D ^(a#5,n−1,1) +D ^(a#5,n−1,2)+D ^(a#5,n−1,3))X _(n−1)(D)+(D ^(b#5,1) +D ^(b#5,2) +D^(b#5,3))P(D)=0  (Equation 7- 5)

(D ^(a#6,1,1) +D ^(a#6,1,2) +D ^(a#6,1,3))X ₁(D)+(D ^(a#6,2,1) +D^(a#6,2,2) +D ^(a#6,2,3))X ₂(D)+ . . . +(D ^(a#6,n−1,1) +D ^(a#6,n−1,2)+D ^(a#6,n−1,3))X _(n−1)(D)+(D ^(b#6,1) +D ^(b#6,2) +D^(b#6,3))P(D)=0  (Equation 7- 6)

At this time, X1(D), X2(D), . . . , Xn−1(D) are polynomialrepresentations of data (information) X1, X2, . . . , Xn−1, and P(D) isa polynomial representation of parity. Here, in equations 7-1 to 7-6,parity check polynomials are assumed such that there are three terms inX1(D), X2(D), . . . , Xn−1(D), and P(D) respectively. As in the case ofthe above coding rate of 1/2, and in the case of a time varying periodof 3, the possibility of being able to obtain higher error correctioncapability is increased if the condition below (<Condition #1>) issatisfied in an LDPC-CC of a time varying period of 6 and a coding rateof (n−1)/n (where n is an integer equal to or greater than 2)represented by parity check polynomials of equations 7-1 to 7-6.

In an LDPC-CC of a time varying period of 6 and a coding rate of (n−1)/n(where n is an integer equal to or greater than 2), parity andinformation at time i are represented by Pi and X_(1,i), X_(1,2), . . ., X_(i,n−1) respectively. If i %6=k (where k=0, 1, 2, 3, 4, 5) isassumed at this time, a parity check polynomial of equation 7-(k+1)holds true. For example, if i=8, i %6=2 (k=2), and therefore equation 8holds true.

[8]

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(8,1)(D)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X _(8,2)(D)+ . . . +(D ^(a#3,n−1,1) +D^(a#3,n−1,2) +D ^(a#3,n−1,3))X _(8,n−1)(D)+(D ^(b#3,1) +D ^(b#3,2) +D^(b#3,3))P ₈=0  (Equation 8)

<Condition #1>

In equations 7-1 to 7-6, combinations of orders of X1(D), X2(D),Xn−1(D), and P(D) satisfy the following condition:

(a _(#1,1,1)%3,a _(#1,1,2)%3,a _(#1,1,3)%3),(a _(#1,2,1)%3,a_(#1,2,2)%3,a _(#1,2,3)%3), . . . ,(a _(#1,k,1)%3,a _(#1,k,2)%3,a_(#1,k,3)%3), . . . ,(a _(#1,n−1,1)%3,a _(#1,n−1,2)%3,a _(#1,n−1,3)%3)and (b _(#1,1)%3,b _(#1,2)%3,b _(#1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where k=1,2,3, . .. ,n−1);

(a _(#2,1,1)%3,a _(#2,1,2)%3,a _(#2,1,3)%3),(a _(#2,2,1)%3,a_(#2,2,2)%3,a _(#2,2,3)%3), . . . ,(a _(#2,k,1)%3,a _(#2,k,2)%3,a_(#2,k,3)%3), . . . ,(a _(#2,n−1,1)%3,a _(#2,n−1,2)%3,a _(#2,n−1,3)%3)and (b _(#2,1)%3,b _(#2,2)%3,b _(#2,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where k=1,2,3, . .. ,n−1);

(a _(#3,1,1)%3,a _(#3,1,2)%3,a _(#3,1,3)%3),(a _(#3,2,1)%3,a_(#3,2,2)%3,a _(#3,2,3)%3), . . . ,(a _(#3,k,1)%3,a _(#3,k,2)%3,a_(#3,k,3)%3), . . . ,(a _(#3,n−1,1)%3,a _(#3,n−1,2)%3,a _(#3,n−1,3)%3)and (b _(#3,1)%3,b _(#3,2)%3,b _(#3,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where k=1,2,3, . .. ,n−1);

(a _(#4,1,1)%3,a _(#4,1,2)%3,a _(#4,1,3)%3),(a _(#4,2,1)%3,a_(#4,2,2)%3,a _(#4,2,3)%3), . . . ,(a _(#4,k,1)%3,a _(#4,k,2)%3,a_(#4,k,3)%3), . . . ,(a _(#4,n−1,1)%3,a _(#4,n−1,2)%3,a _(#4,n−1,3)%3)and (b _(#4,1)%3,b _(#4,2)%3,b _(#4,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where k=1,2,3, . .. ,n−1);

(a _(#5,1,1)%3,a _(#5,1,2)%3,a _(#5,1,3)%3),(a _(#5,2,1)%3,a_(#5,2,2)%3,a _(#5,2,3)%3), . . . ,(a _(#5,k,1)%3,a _(#5,k,2)%3,a_(#5,k,3)%3), . . . ,(a _(#5,n−1,1)%3,a _(#5,n−1,2)%3,a _(#5,n−1,3)%3)and (b _(#5,1)%3,b _(#5,2)%3,b _(#5,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where k=1,2,3, . .. ,n−1); and

(a _(#6,1,1)%3,a _(#6,1,2)%3,a _(#6,1,3)%3),(a _(#6,2,1)%3,a_(#6,2,2)%3,a _(#6,2,3)%3), . . . ,(a _(#6,k,1)%3,a _(#6,k,2)%3,a_(#6,k,3)%3), . . . ,(a _(#6,n−1,1)%3,a _(#6,n−1,2)%3,a _(#6,n−1,3)%3)and (b _(#6,1)%3,b _(#6,2)%3,b _(#6,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where k=1,2,3, . .. ,n−1).

In the above description, a code having high error correction capabilityhas been described for an LDPC-CC of a time varying period of 6, but acode having high error correction capability can also be generated whenan LDPC-CC of a time varying period of 3g (where g=1, 2, 3, 4, . . . )(that is, an LDPC-CC for which the time varying period is a multiple of3) is created in the same way as with the design method for an LDPC-CCof a time varying period of 3 or 6. A configuration method for this codeis described in detail below.

Consider equations 9-1 to 9-3g as parity check polynomials of an LDPC-CCfor which the time varying period is 3g (where g=1, 2, 3, 4, . . . ) andthe coding rate is (n−1)/n (where n is an integer equal to or greaterthan 2).

[9]

(D ^(a#1,1,1) +D ^(a#1,1,2) +D ^(a#1,1,3))X ₁(D)+(D ^(a#1,2,1) +D^(a#1,2,2) +D ^(a#1,2,3))X ₂(D)+ . . . +(D ^(a#1,n−1,1) +D ^(a#1,n−1,2)+D ^(a#1,n−1,3))X _(n−1)(D)+(D ^(b#1,1) +D ^(b#1,2) +D^(b#1,3))P(D)=0  (Equation 9- 1)

(D ^(a#2,1,1) +D ^(a#2,1,2) +D ^(a#2,1,3))X ₁(D)+(D ^(a#2,2,1) +D^(a#2,2,2) +D ^(a#2,2,3))X ₂(D)+ . . . +(D ^(a#2,n−1,1) +D ^(a#2,n−1,2)+D ^(a#2,n−1,3))X _(n−1)(D)+(D ^(b#2,1) +D ^(b#2,2) +D^(b#2,3))P(D)=0  (Equation 9- 2)

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X ₁(D)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X ₂(D)+ . . . +(D ^(a#3,n−1,1) +D ^(a#3,n−1,2)+D ^(a#3,n−1,3))X _(n−1)(D)+(D ^(b#3,1) +D ^(b#3,2) +D^(b#3,3))P(D)=0  (Equation 9- 3)

(D ^(a#k,1,1) +D ^(a#k,1,2) +D ^(a#k,1,3))X ₁(D)+(D ^(a#k,2,1) +D^(a#k,2,2) +D ^(a#k,2,3))X ₂(D)+ . . . +(D ^(a#k,n−1,1) +D ^(a#k,n−1,2)+D ^(a#k,n−1,3))X _(n−1)(D)+(D ^(b#k,1) +D ^(b#k,2) +D^(b#k,3))P(D)=0  (Equation 9- k)

(D ^(a#3g−2,1,1) +D ^(a#3g−2,1,2) +D ^(a#3g−2,1,3))X ₁(D)+(D^(a#3g−2,2,1) +D ^(a#3g−2,2,2) +D ^(a#3g−2,2,3))X ₂(D)+ . . . +(D^(a#3g−2,n−1,1) +D ^(a#3g−2,n−1,2) +D ^(a#3g−2,n−1,3))X _(n−1)(D)+(D^(b#3g−2,1) +D ^(b#3g−2,2) +D ^(b#3g−2,3))P(D)=0  (Equation 9-(3g−2))

(D ^(a#3g−1,1,1) +D ^(a#3g−1,1,2) +D ^(a#3g−1,1,3))X ₁(D)+(D^(a#3g−1,2,1) +D ^(a#3g−1,2,2) +D ^(a#3g−1,2,3))X ₂(D)+ . . . +(D^(a#3g−1,n−1,1) +D ^(a#3g−1,n−1,2) +D ^(a#3g−1,n−1,3))X _(n−1)(D)+(D^(b#3g−1,1) +D ^(b#3g−1,2) +D ^(b#3g−1,3))P(D)=0  (Equation 9-(3g−1))

(D ^(a#3g,1,1) +D ^(a#3g,1,2) +D ^(a#3g,1,3))X ₁(D)+(D ^(a#3g,2,1) +D^(a#3g,2,2) +D ^(a#3g,2,3))X ₂(D)+ . . . +(D ^(a#3g,n−1,1) +D^(a#3g,n−1,2) +D ^(a#3g,n−1,3))X _(n−1)(D)+(D ^(b#3g,1) +D ^(b#3g,2) +D^(b#3g,3))P(D)=0  (Equation 9-3g)

At this time, X1(D), X2(D), . . . , Xn−1(D) are polynomialrepresentations of data (information) X1, X2, . . . , Xn−1, and P(D) isa polynomial representation of parity. Here, in equations 9-1 to 9-3g,parity check polynomials are assumed such that there are three terms inX1(D), X2(D), . . . , Xn−1(D), and P(D) respectively.

As in the case of an LDPC-CC of a time varying period of 3 and anLDPC-CC of a time varying period of 6, the possibility of being able toobtain higher error correction capability is increased if the conditionbelow (Condition #2>) is satisfied in an LDPC-CC of a time varyingperiod of 3g and a coding rate of (n−1)/n (where n is an integer equalto or greater than 2) represented by parity check polynomials ofequations 9-1 to 9-3g.

In an LDPC-CC of a time varying period of 3g and a coding rate of(n−1)/n (where n is an integer equal to or greater than 2), parity andinformation at time i are represented by P_(i) and X_(i,1), X_(i,2), . .. , X_(i,n−1) respectively. If i %3g=k (where k=0, 1, 2, . . . , 3g−1)is assumed at this time, a parity check polynomial of equation 9-(k+1)holds true. For example, if i=2, i %3g=2 (k=2), and therefore equation10 holds true.

[10]

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(2,1)(D)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X _(2,2)(D)+ . . . +(D ^(a#3,n−1,1) +D^(a#3,n−1,2) +D ^(a#3,n−1,3))X _(2,n−1)(D)+(D ^(b#3,1) +D ^(b#3,2) +D^(b#3,3))P ₂=0  (Equation 10)

In equations 9-1 to 9-3g, it is assumed that a_(#k,p,1), a_(#k,p,2) anda_(#k,p,3) are integers (where a_(#k,p,1)≠a_(#k,p,2)≠a_(#k,p,3)) (wherek=1, 2, 3, . . . , 3g, and p=1, 2, 3, . . . , n−1). Also, it is assumedthat b_(#k,1), b_(#k,2) and b_(#k,3) are integers (whereb_(#k,1)≠b_(#k,2)≠b_(#k,3)). A parity check polynomial of equation 9-k(where k=1, 2, 3, . . . , 3g) is called “check equation #k,” and asub-matrix based on the parity check polynomial of equation 9-k isdesignated k-th sub-matrix H_(k). Next, an LDPC-CC of a time varyingperiod of 3g is considered that is generated from first sub-matrix H₁,second sub-matrix H₂, third sub-matrix H₃, . . . , and 3g-th sub-matrixH_(3g).

<Condition #2>

In equations 9-1 to 9-3g, combinations of orders of X1(D), X2(D),Xn−1(D), and P(D) satisfy the following condition:

(a _(#1,1,1)%3,a _(#1,1,2)%3,a _(#1,1,3)%3),(a _(#1,2,1)%3,a_(#1,2,2)%3,a _(#1,2,3)%3), . . . ,(a _(#1,p,1)%3,a _(#1,p,2)%3,a_(#1,p,3)%3), . . . ,(a _(#1,n−1,1)%3,a _(#1,n−1,2)%3,a _(#1,n−1,3)%3)and (b _(#1,1)%3,b _(#1,2)%3,b _(#1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where p=1,2,3, . .. ,n−1);

(a _(#2,1,1)%3,a _(#2,1,2)%3,a _(#2,1,3)%3),(a _(#2,2,1)%3,a_(#2,2,2)%3,a _(#2,2,3)%3), . . . ,(a _(#2,p,1)%3,a _(#2,p,2)%3,a_(#2,p,3)%3), . . . ,(a _(#2,n−1,1)%3,a _(#2,n−1,2)%3,a _(#2,n−1,3)%3)and (b _(#2,1)%3,b _(#2,2)%3,b _(#2,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where p=1,2,3, . .. ,n−1);

(a _(#3,1,1)%3,a _(#3,1,2)%3,a _(#3,1,3)%3),(a _(#3,2,1)%3,a_(#3,2,2)%3,a _(#3,2,3)%3), . . . ,(a _(#3,p,1)%3,a _(#3,p,2)%3,a_(#3,p,3)%3),(a _(#3,n−1),1%3,a _(#3,n−1),2%3,a _(#3,n−1),3%3) and (b_(#3,1)%3,b _(#3,2)%3,b _(#3,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where p=1,2,3, . .. ,n−1);

(a _(#k,1,1)%3,a _(#k,1,2)%3,a _(#k,1,3)%3),(a _(#k,2,1)%3,a_(#k,2,2)%3,a _(#k,2,3)%3), . . . ,(a _(#k,p,1)%3,a _(#k,p,2)%3,a_(#k,p,3)%3), . . . ,(a _(#k,n−1,1)%3,a _(#k,n−1,2)%3,a _(#k,n−1,3)%3)and (b _(#k,1)%3,b _(#k,2)%3,b _(#k,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where p=1,2,3, . .. ,n−1) (where,k=1,2,3, . . . ,3g);

(a _(#3g−2,1,1)%3,a _(#3g−2,1,2)%3,a _(#3g−2,1,3)%3),(a _(#3g−2,2,1)%3,a_(#3g−2,2,2)%3,a _(#3g−2,2,3)%3), . . . ,(a _(#3g−2,p,1)%3,a_(#3g−2,p,2)%3,a _(#3g−2,p,3)%3), . . . ,(a _(#3g−2,n−1,1)%3,a_(#3g−2,n−1,2)%3,a _(#3g−2,n−1,3)%3), and (b _(#3g−2,1)%3,b_(#3g−2,2)%3,b _(#3g−2,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where p=1,2,3, . .. ,n−1);

(a _(#3g−1,1,1)%3,a _(#3g−1,1,2)%3,a _(#3g−1,1,3)%3),(a _(#3g−1,2,1)%3,a_(#3g−1,2,2)%3,a _(#3g−1,2,3)%3), . . . ,(a _(#3g−1,p,1)%3,a_(#3g−1,p,2)%3,a _(#3g−1,p,3)%3), . . . ,(a _(#3g−1,n−1,1)%3,a_(#3g−1,n−1,2)%3,a _(#3g−1,n−1,3)%3) and (b _(#3g−1,1)%3,b_(#3g−1,2)%3,b _(#3g−1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where p=1,2,3, . .. ,n−1); and

(a _(#3g,1,1)%3,a _(#3g,1,2)%3,a _(#3g,1,3)%3),(a _(#3g,2,1)%3,a_(#3g,2,2)%3,a _(#3g,2,3)%3), . . . ,(a _(#3g,p,1)%3,a _(#3g,p,2)%3,a_(#3g,p,3)%3), . . . ,(a _(#3g,n−1,1)%3,a _(#3g,n−1,2)%3,a_(#3g,n−1,3)%3) and (b _(#3g,1)%3,b _(#3g,2)%3,b _(#3g,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where p=1,2,3, . .. ,n−1).

Taking ease of performing encoding into consideration, it is desirablefor one “0” to be present among the three items (b_(#10)%3, b_(#k,2)%3,b_(#k,3)%3) (where k=1, 2, . . . , 3g) in equations 9-1 to 9-3g. This isbecause of a feature that, if D⁰=1 holds true and b_(#k,1), b_(#k,2) andb_(#k,3) are integers equal to or greater than 0 at this time, parity Pcan be found sequentially.

Also, in order to provide relevancy between parity bits and data bits ofthe same point in time, and to facilitate a search for a code havinghigh correction capability, it is desirable for:

one “0” to be present among the three items (a_(#k,1,1)%3, a_(#k,1,2)%3,a_(#k,1,3)%3);

one “0” to be present among the three items (a_(#k,2,1)%3, a_(#k,2,2)%3,a_(#k,2,3)%3);

-   -   

one “0” to be present among the three items (a_(#k,p,1)%3, a_(#k,p,2)%3,a_(#k,p,3)%3);

-   -   

one “0” to be present among the three items (a_(#k,n−1,1)%3,a_(#k, n−1,2)%3, a_(#k, n−1,3)%3), (where k=1, 2, . . . , 3g).

Next, an LDPC-CC of a time varying period of 3g (where g=2, 3, 4, 5, . .. ) that takes ease of encoding into account is considered. At thistime, if the coding rate is (n−1)/n (where n is an integer equal to orgreater than 2), LDPC-CC parity check polynomials can be represented asshown below.

[11]

(D ^(a#1,1,1) +D ^(a#1,1,2) +D ^(a#1,1,3))X ₁(D)+(D ^(a#1,2,1) +D^(a#1,2,2) +D ^(a#1,2,3))X ₂(D)+ . . . +(D ^(a#1,n−1,1) +D ^(a#1,n−1,2)+D ^(a#1,n−1,3))X _(n−1)(D)+(D ^(b#1,1) +D ^(b#1,2) +D^(b#1,3))P(D)=0  (Equation 11-1)

(D ^(a#2,1,1) +D ^(a#2,1,2) +D ^(a#2,1,3))X ₁(D)+(D ^(a#2,2,1) +D^(a#2,2,2) +D ^(a#2,2,3))X ₂(D)+ . . . +(D ^(a#2,n−1,1) +D ^(a#2,n−1,2)+D ^(a#2,n−1,3))X _(n−1)(D)+(D ^(b#2,1) +D ^(b#2,2) +D^(b#2,3))P(D)=0  (Equation 11-2)

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X ₁(D)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X ₂(D)+ . . . +(D ^(a#3,n−1,1) +D ^(a#3,n−1,2)+D ^(a#3,n−1,3))X _(n−1)(D)+(D ^(b#3,1) +D ^(b#3,2) +D^(b#3,3))P(D)=0  (Equation 11-3)

(D ^(a#k,1,1) +D ^(a#k,1,2) +D ^(a#k,1,3))X ₁(D)+(D ^(a#k,2,1) +D^(a#k,2,2) +D ^(a#k,2,3))X ₂(D)+ . . . +(D ^(a#k,n−1,1) +D ^(a#k,n−1,2)+D ^(a#k,n−1,3))X _(n−1)(D)+(D ^(b#k,1) +D ^(b#k,2) +D^(b#k,3))P(D)=0  (Equation 11-k)

(D ^(a#3g−2,1,1) +D ^(a#3g−2,1,2) +D ^(a#3g−2,1,3))X ₁(D)+(D^(a#3g−2,2,1) +D ^(a#3g−2,2,2) +D ^(a#3g−2,2,3))X ₂(D)+ . . . +(D^(a#3g−2,n−1,1) +D ^(a#3g−2,n−1,2) +D ^(a#3g−2,n−1,3))X _(n−1)(D)+(D^(b#3g−2,1) +D ^(b#3g−2,2) +D ^(b#3g−2,3))P(D)=0  (Equation 11-(3g−2))

(D ^(a#3g−1,1,1) +D ^(a#3g−1,1,2) +D ^(a#3g−1,1,3))X ₁(D)+(D^(a#3g−1,2,1) +D ^(a#3g−1,2,2) +D ^(a#3g−1,2,3))X ₂(D)+ . . . +(D^(a#3g−1,n−1,1) +D ^(a#3g−1,n−1,2) +D ^(a#3g−1,n−1,3))X _(n−1)(D)+(D^(b#3g−1,1) +D ^(b#3g−1,2) +D ^(b#3g−1,3))P(D)=0  (Equation 11-(3g−1))

(D ^(a#3g,1,1) +D ^(a#3g,1,2) +D ^(a#3g,1,3))X ₁(D)+(D ^(a#3g,2,1) +D^(a#3g,2,2) +D ^(a#3g,2,3))X ₂(D)+ . . . +(D ^(a#3g,n−1,1) +D^(a#3g,n−1,2) +D ^(a#3g,n−1,3))X _(n−1)(D)+(D ^(b#3g,1) +D ^(b#3g,2) +D^(b#3g,3))P(D)=0  (Equation 11-3g)

At this time, X1(D), X2(D), . . . , Xn−1(D) are polynomialrepresentations of data (information) X1, X2, . . . , Xn−1, and P(D) isa polynomial representation of parity. Here, in equations 11-1 to 11-3g,parity check polynomials are assumed such that there are three terms inX1(D), X2(D), . . . , Xn−1(D), and P(D) respectively. In an LDPC-CC of atime varying period of 3g and a coding rate of (n−1)/n (where n is aninteger equal to or greater than 2), parity and information at time iare represented by Pi and X_(i,1), X_(i,2), . . . , X_(i,n−1)respectively. If i %3g=k (where k=0, 1, 2, . . . , 3g−1) is assumed atthis time, a parity check polynomial of equation 11-(k+1) holds true.For example, if i=2, i %3=2 (k=2), and therefore equation 12 holds true.

[12]

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(2,1)+(D ^(a#3,2,1) +D^(a#3,2,2) +D ^(a#3,2,3))X _(2,2)+ . . . +(D ^(a#3,n−1,1) +D^(a#3,n−1,2) +D ^(a#3,n−1,3))X _(2,n−1)+(D ^(b#3,1) +D^(b#3,2)+1)P(D)=0  (Equation 12)

If <Condition #3> and <Condition #4> are satisfied at this time, thepossibility of being able to create a code having higher errorcorrection capability is increased.

<Condition #3>

In equations 11-1 to 11-3g, combinations of orders of X1(D), X2(D), . .. , Xn−1(D), and P(D) satisfy the following condition:

(a _(#1,1,1)%3,a _(#1,1,2)%3,a _(#1,1,3)%3),(a _(#1,2,1)%3,a_(#1,2,2)%3,a _(#1,2,3)%3), . . . ,(a _(#1,p,1)%3,a _(#1,p,2)%3,a_(#1,p,3)%3), . . . , and (a _(#1,n−1,1)%3,a _(#1,n−1,2)%3,a_(#1,n−1,3)%3) are any of (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or(2,1,0) (where p=1,2,3, . . . ,n−1);

(a _(#2,1,1)%3,a _(#2,1,2)%3,a _(#2,1,3)%3),(a _(#2,2,1)%3,a_(#2,2,2)%3,a _(#2,2,3)%3), . . . ,(a _(#4,1)%3,a _(#2,p,2)%3,a_(#2,p,3)%3), . . . , and (a _(#2,n−1,1)%3,a _(#2,n−1,2)%3,a_(#2,n−1,3)%3) are any of (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or(2,1,0) (where p=1,2,3, . . . ,n−1);

(a _(#3,1,1)%3,a _(#312)%3,a _(#313)%3),(a _(#3,2,1)%3,a _(#3,2,2)%3,a_(#3,2,3)%3), . . . ,(a _(#3,p,1)%3,a _(#3,p,2)%3,a _(#3,p,3)%3), . . ., and (a _(#3,n−1,1)%3,a _(#3,n−1,2)%3,a _(#3,n−1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where p=1,2,3, . .. ,n−1);

(a _(#k,1,1)%3,a _(#k,1,2)%3,a _(#k,1,3)%3),(a _(#k,2,1)%3,a_(#k,2,2)%3,a _(#k,2,3)%3), . . . ,(a _(#k,p,1)%3,a _(#k,p,2)%3,a_(#k,p,3)%3), . . . , and (a _(#k,n−1,1)%3,a _(#k,n−1,2)%3,a_(#k,n−1,3)%3) are any of (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or(2,1,0) (where p=1,2,3, . . . ,n−1, and k=1,2,3, . . . ,3g);

(a _(#3g−2,1,1)%3,a _(#3g−2,1,2)%3,a _(#3g−2,1,3)%3),(a _(#3g−2,2,1)%3,a_(#3g−2,2,2)%3,a _(#3g−2,2,3)%3),(a _(#3g−2,p,1)%3,a _(#3g−2,p,2)%3,a_(#3g−2,p,3)%3), . . . , and (a _(#3g−2,n−1,1)%3,a _(#3g−2,n−1,2)%3,a_(#3g−2,n−1,3)%3) are any of (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or(2,1,0) (where p=1,2,3, . . . ,n−1);

(a _(#3g−1,1,1)%3,a _(#3g−1,1,2)%3,a _(#3g,1,3)%3),(a _(#3g−1,2,1)%3,a_(#3g−1,2,2)%3,a _(#3g−1,2,3)%3),(a _(#3g−1,p,1)%3,a _(#3g−1,p,2)%3,a_(#3g−1,p,3)%3), . . . , and (a _(#3g−1,n−1,1)%3,a _(#3g−1,n−1,2)%3,a_(#3g−1,n−1,3)%3) are any of (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or(2,1,0) (where p=1,2,3, . . . ,n−1); and

(a _(#3g,1)%3,a _(#3g,1,2)%3,a _(#3g,1,3)%3),(a _(#3g,2,1)%3,a_(#3g,2,2)%3,a _(#3g,2,3)%3),(a _(#3g,p,1)%3,a _(#3g,p,2)%3,a_(#3g,p,3)%3), . . . , and (a _(#3g,n−1,1)%3,a _(#3g,n−1,2)%3,a_(#3g,n−1,3)%3) are any of (0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or(2,1,0) (where p=1,2,3, . . . ,n−1).

In addition, in equations 11-1 to 11-3g, combinations of orders of P(D)satisfy the following condition:

(b _(#1,1)%3,b _(#1,2)%3),(b _(#2,1)%3,b _(#2,2)%3),(b _(#3,1)%3,b_(#3,2)%3),(b _(#k,1)%3,b _(#k,2)%3),(b _(#3g−2,1)%3,b _(#3g−2,2)%3),(b_(#3g−1,1)%3,b _(#3g−1,2)%3), and (b _(#3g,1)%3,b _(#3g,2)%3) are any of(1,2), or (2,1) (where k=1,2,3, . . . ,3g).

<Condition #3> has a similar relationship with respect to equations 11-1to 11-3g as <Condition #2> has with respect to equations 9-1 to 9-3g. Ifthe condition below (<Condition #4>) is added for equations 11-1 to11-3g in addition to <Condition #3>, the possibility of being able tocreate an LDPC-CC having higher error correction capability isincreased.

<Condition #4>

Orders of P(D) of equations 11-1 to 11-3g satisfy the followingcondition:

all values other than multiples of 3 (that is, 0, 3, 6, . . . , 3g−3)from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2, 3g−1)are present in the values of 6g orders of (b_(#1,1)%3g, b_(#1,2)%3g),(b_(#2,1)%3g, b_(#2,2)%3g), (b_(#3,1)%3g, b_(#3,2)%3g), . . . ,(b_(#k,1)%3g, b_(#k,2)%3g), . . . , (b_(#3g)- 2,1%3g, b_(#3g−2,2)%3g),(b_(#3g−1,1)%3g, b_(#3g−1,2)%3g), (b_(#3g,1)%3g, b_(#3g,2)%3g) (in thiscase, two orders form a pair, and therefore the number of orders forming3g pairs is 6g).

The possibility of obtaining good error correction capability is high ifthere is also randomness while regularity is maintained for positions atwhich “1”s are present in a parity check matrix. With an LDPC-CC forwhich the time varying period is 3g (where g=2, 3, 4, 5, . . . ) and thecoding rate is (n−1)/n (where n is an integer equal to or greater than2) that has parity check polynomials of equations 11-1 to 11-3g, if acode is created in which <Condition #4> is applied in addition to<Condition #3>, it is possible to provide randomness while maintainingregularity for positions at which “1”s are present in a parity checkmatrix, and therefore the possibility of obtaining good error correctioncapability is increased.

Next, an LDPC-CC of a time varying period of 3g (where g=2, 3, 4, 5, . .. ) is considered that enables encoding to be performed easily andprovides relevancy to parity bits and data bits of the same point intime. At this time, if the coding rate is (n−1)/n (where n is an integerequal to or greater than 2), LDPC-CC parity check polynomials can berepresented as shown below.

[13]

(D ^(a#1,1,1) +D ^(a#1,1,2)+1)X ₁(D)+(D ^(a#1,2,1) +D ^(a#1,2,2)+1)X₂(D)+ . . . +(D ^(a#1,n−1,1) +D ^(a#1,n−1,2)+1)X _(n−1)(D)+(D ^(b#1,1)+D ^(b#1,2)+1)P(D)=0  (Equation 13-1)

(D ^(a#2,1,1) +D ^(a#2,1,2)+1)X ₁(D)+(D ^(a#2,2,1) +D ^(a#2,2,2)1)X₂(D)+ . . . +(D ^(a#2,n−1,1) +D ^(a#2,n−1,2)+1)X _(n−1)(D)+(D ^(b#2,1)+D ^(b#2,2)+1)P(D)=0  (Equation 13-2)

(D ^(a#3,1,1) +D ^(a#3,1,2)+1)X ₁(D)+(D ^(a#3,2,1) +D ^(a#3,2,2)+1)X₂(D)+ . . . +(D ^(a#3,n−1,1) +D ^(a#3,n−1,2)+1)X _(n−1)(D)+(D ^(b#3,1)+D ^(b#3,2)+1)P(D)=0  (Equation 13-3)

(D ^(a#k,1,1) +D ^(a#k,1,2)+1)X ₁(D)+(D ^(a#k,2,1) +D ^(a#k,2,2)+1)X₂(D)+ . . . +(D ^(a#k,n−1,1) +D ^(a#k,n−1,2)+1)X _(n−1)(D)+(D ^(b#k,1)+D ^(b#k,2)+1)P(D)=0  (Equation 13-k)

(D ^(a#3g−2,1,1) +D ^(a#3g−2,1,2)+1)X ₁(D)+(D ^(a#3g−2,2,1) +D^(a#3g−2,2,2)+1)X ₂(D)+ . . . +(D ^(a#3g−2,n−1,1) +D ^(a#3g−2,n−1,2)+1)X_(n−1)(D)+(D ^(b#3g−2,1) +D ^(b#3g−2,2)+1)P(D)=0  (Equation 13-(3g−2))

(D ^(a#3g−1,1,1) +D ^(a#3g−1,1,2)+1)X ₁(D)+(D ^(a#3g−1,2,1) +D^(a#3g−1,2,2)+1)X ₂(D)+ . . . +(D ^(a#3g−1,n−1,1) +D ^(a#3g−1,n−1,2)+1)X_(n−1)(D)+(D ^(b#3g−1,1) +D ^(b#3g−1,2)+1)P(D)=0  (Equation 13-(3g−1))

(D ^(a#3g,1,1) +D ^(a#3g,1,2)+1)X ₁(D)+(D ^(a#3g,2,1) +D ^(a#3g,2,2)+1)X₂(D)+ . . . +(D ^(a#3g,n−1,1) +D ^(a#3g,n−1,2)+1)X _(n−1)(D)+(D^(b#3g,1) +D ^(b#3g,2)+1)P(D)=0  (Equation 13-3g)

At this time, X1(D), X2(D), . . . , Xn−1(D) are polynomialrepresentations of data (information) X1, X2, . . . , Xn−1, and P(D) isa polynomial representation of parity. In equations 13-1 to 13-3g,parity check polynomials are assumed such that there are three terms inX1(D), X2(D), . . . , Xn−1(D), and P(D) respectively, and term D⁰ ispresent in X1(D), X2(D), . . . , Xn−1(D), and P(D) (where k=1, 2, 3, . .. , 3g).

In an LDPC-CC of a time varying period of 3g and a coding rate of(n−1)/n (where n is an integer equal to or greater than 2), parity andinformation at time i are represented by Pi and X_(1,i), X_(1,2), . . ., X_(i,n−1) respectively. If i %3g=k (where k=0, 1, 2, . . . 3g−1) isassumed at this time, a parity check polynomial of equation 13-(k+1)holds true. For example, if i=2, i %3g=2 (k=2), and therefore equation14 holds true.

[14]

(D ^(a#3,1,1) +D ^(a#3,1,2)+1)X _(2,1)(D)+(D ^(a#3,2,1) +D^(a#3,2,2)+1)X _(2,2)(D)+ . . . +(D ^(a#3,n−1,1) +D ^(a#3,n−1,2)+1)X_(2,n−1)+(D ^(b#3,1) +D ^(b#3,2)+1)P ₂=0  (Equation 13-3)

If following <Condition #5> and <Condition #6> are satisfied at thistime, the possibility of being able to create a code having higher errorcorrection capability is increased.

<Condition #5>

In equations 13-1 to 13-3g, combinations of orders of X1(D), X2(D),Xn−1(D), and P(D) satisfy the following condition:

(a _(#1,1,1)%3,a _(#1,1,2)%3),(a _(#1,2,1)%3,a _(#1,2,2)%3), . . . ,(a_(#1,p,1)%3,a _(#1,p,2)%3), . . . , and (a _(#1,n−1,1)%3,a_(#1,n−1,2)%3) are any of (1,2),(2,1) (p=1,2,3, . . . ,n−1);

(a _(#2,1,1)%3,a _(#2,1,2)%3),(a _(#2,2,1)%3,a _(#2,2,2)%3), . . . ,(a_(#2,p,1)%3,a _(#2,p,2)%3), . . . , and (a _(#2,n−1)%3,a _(#2,n−1,2)%3)are any of (1,2), or (2,1) (where p=1,2,3, . . . ,n−1);

(a _(#3,1,1)%3,a _(#3,1,2)%3),(a _(#3,2,1)%3,a _(#3,2,2)%3), . . . ,(a_(#3,p,1)%3,a _(#3,p,2)%3), . . . , and (a _(#3n−1,1)%3,a _(#3,n−1,2)%3)are any of (1,2), or (2,1) (where p=1,2,3, . . . ,n−1);

(a _(#k,1,1)%3,a _(#k,1,2)%3),(a _(#k,2,1)%3,a _(#k,2,2)%3),(a_(#k,p,1)%3,a _(#k,p,2)%3), . . . , and (a _(#k,n−1,1)%3,a_(#k,n−1,2)%3) are any of (1,2), or (2,1) (where p=1,2,3, . . . ,n−1)(where,k=1,2,3, . . . ,3g)

(a _(#3g−2,1,1)%3,a _(#3g−2,1,2)%3),(a _(#3g−2,2,1)%3,a _(#3g−2,2,2)%3),. . . ,(a _(#3g−2,p,1)%3,a _(#3g−2,p,2)%3), . . . , and (a_(#3g−2,n−1,1)%3,a _(#3g−2,n−1,2)%3) are any of (1,2), or (2,1) (wherep=1,2,3, . . . ,n−1);

(a _(#3g−1,1,1)%3,a _(#3g−1,1,2)%3),(a _(#3g−1,2,1)%3,a _(#3g−1,2,2)%3),. . . ,(a _(#3g−1,p,1)%3,a _(#3g−1,p,2)%3), . . . , and (a_(#3g−1,n−1,1)%3,a _(#3g−1,n−1,2)%3) are any of (1,2), or (2,1) (wherep=1,2,3, . . . ,n−1); and

(a _(#3g,1,1)%3,a _(#3g,1,2)%3),(a _(#3g,2,1)%3,a _(#3g,2,2)%3), . . .,(a _(#3g,p,1)%3,a _(#3g,p,2)%3), . . . , and (a _(#3g,n−1,1)%3,a_(#3g,n−1,2)%3) are any of (1,2), or (2,1) (where p=1,2,3, . . . ,n−1).

In addition, in equations 13-1 to 13-3g, combinations of orders of P(D)satisfy the following condition:

(b _(#1,1)%3,b _(#1,2)%3),(b _(#2,1)%3,b _(#2,2)%3),(b _(#3,1)%3,b_(#3,2)%3), . . . ,(b _(#k,1)%3,b _(#k,2)%3), . . . ,(b _(#3g−2,1)%3,b_(#3g−2,2)%3),(b _(#3g−1,1)%3,b _(#3g−1,2)%3), and (b _(#3g,1)%3,b_(#3g,2)%3) are any of (1,2), or (2,1) (where k=1,2,3, . . . ,3g).

<Condition #5> has a similar relationship with respect to equations 13-1to 13-3g as <Condition #2> has with respect to equations 9-1 to 9-3g. Ifthe condition below (<Condition #6>) is added for equations 13-1 to13-3g in addition to <Condition #5>, the possibility of being able tocreate a code having high error correction capability is increased.

<Condition #6>

Orders of X1 (D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,1,1)%3g,a_(#1,1,2)%3g), (a_(#2,1,1)%3g), . . . , a_(#2,1,2)%3g, . . . ,(a_(#p,1,1)%3g, a_(#p,1,2)%3g), . . . , and (a_(#3g,1,1)%3g,a_(#3g,1,2)%3g) (where p=1, 2, 3, . . . , 3g);

Orders of X2(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,2,1)%3g,a_(#1,2,2)%3g), (a_(#2,2,1)%3g, a_(#2,2,2)%3g), . . . , (a_(#p,2,1)%3g,a_(#p,2,2)%3g), . . . , and (a_(#3g,2,1)%3g, a_(#3g,2,2)%3g) (where p=1,2, 3, . . . , 3g);

Orders of X3(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,3,1)%3g,a_(#1,3,2)%3g), (a_(#2,3,1)%3g, a_(#2,3,2)%3g), . . . , (a_(#p,3,1)%3g,a_(#p,3,2)%3g), . . . , and (a_(#3g,3,1)%3g, a_(#3g,3,2)%3g) (where p=1,2, 3, . . . , 3g);

-   -   

Orders of Xk(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,k,1)%3g,a_(#1,k,2)%3g), (a_(#2,k,1)%3g, a_(#2,k,2)%3g), . . . , (a_(#p,k,1)%3g,a_(#p,k,2)%3g), . . . , and (a_(#3g,k),1%3g, a_(#3g,k),2%3g) (where p=1,2, 3, . . . , 3g, and k=1, 2, 3, . . . , n−1);

-   -   

Orders of Xn−1(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,n−1,1)%3g,a_(#1,n−1,2)%3g), (a_(#2,n−1,1)%3g, a_(#2,n−1,2)%3g), (a_(#p,n−1,1)%3g,a_(#p,n−1,2)%3g), . . . , and (a_(#3g,n−1,1)%3g, a_(#3g,n−1,2)%3g)(where p=1, 2, 3, . . . , 3g); and

Orders of P(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (b_(#1,1)%3g,b_(#1,2)%3g), (b_(#2,1)%3g, b_(#2,2)%3g), (b_(#3,1)%3g, b_(#3,2)%3g),(b_(#k,1)%3g, b_(#k,2)%3g), (b_(#3g−2,1)%3g, b_(#3g−2,2)%3g),(b_(#3g−1,1)%3g, b_(#3g−1,2)%3g) and (b_(#3g,1)%3g, b_(#3g,2)%3g) (wherek=1, 2, 3, . . . , n−1).

The possibility of obtaining good error correction capability is high ifthere is also randomness while regularity is maintained for positions atwhich “1”s are present in a parity check matrix. With an LDPC-CC forwhich the time varying period is 3g (where g=2, 3, 4, 5, . . . ) and thecoding rate is (n−1)/n (where n is an integer equal to or greater than2) that has parity check polynomials of equations 13-1 to 13-3g, if acode is created in which <Condition #6> is applied in addition to<Condition #5>, it is possible to provide randomness while maintainingregularity for positions at which “1”s are present in a parity checkmatrix, and therefore the possibility of obtaining good error correctioncapability is increased.

The possibility of being able to create an LDPC-CC having higher errorcorrection capability is also increased if a code is created using<Condition #6′> instead of <Condition #6>, that is, using <Condition#6′> in addition to <Condition #5>.

<Condition #6′>

Orders of X1(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,1,1)%3g,a_(#1,1,2)%3g), (a_(#2,1,1)%3g, a_(#2,1,2)%3g), . . . , (a_(#p,1,1)%3g,a_(#p,1,2)%3g), . . . , and (a_(#3g,1)%3g, a_(#3g,1,2)%3g) (where p=1,2, 3, . . . , 3g);

Orders of X2(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,2,1)%3g,a_(#1,2,2)%3g), (a_(#2,2,1)%3g, a_(#2,2,2)%3g), . . . , (a_(#p,2,1)%3g,a_(#p,2,2)%3g), . . . , and (a_(#3g,2,1)%3g, a_(#3g,2,2)%3 g) (wherep=1, 2, 3, . . . , 3g);

Orders of X3(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,3,1)%3g,a_(#1,3,2)%3g), (a_(#2,3,1)%3g, a_(#2,3,2)%3g), . . . , (a_(#p,3,1)%3g,a_(#p,3,2)%3g), . . . , and (a_(#3g,3,1)%3g, a_(#3g,3,2)%3g) (where p=1,2, 3, . . . , 3g);

-   -   

Orders of Xk(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,k,1)%3g,a_(#1,k,2)%3g), (a_(#2,k,1)%3g, a_(#2,k,2)%3g), . . . , (a_(#p,k,1)%3g,a_(#p,k,2)%3g), . . . , (a_(#3g,k,1)%3g, a_(#3g,k,2)%3g) (where p=1, 2,3, . . . , 3g, and k=1, 2, 3, . . . , n−1);

-   -   

Orders of Xn−1(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,n−1,1)%3g,a_(#1,n−1,2)%3g), (a_(#2,n−1,1)%3g, a_(#2,n−1,2)%3g), . . . ,(a_(#p,n−1,1)%3g, a_(#p,n−1,2)%3g), . . . , (a_(#3g,n−1,1)%3g,a_(#3g,n−1,2)%3g) (where p=1, 2, 3, . . . , 3g); or

Orders of P(D) of equations 13-1 to 13-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (b_(#1,1)%3g,b_(#1,2)%3g), (b_(#2,1)%3g, b_(#2,2)%3g), (b_(#3,1)%3g, b_(#3,2)%3g), .. . , (b_(#k,1)%3g, b_(#k,2)%3g), . . . , (b_(#3g−2,1)%3g,b_(#3g−2,2)%3g), (b_(#3g−1,1)%3g, b_(#3g−1,2)%3g), (b_(#3g,1)%3g,b_(#3g,2)%3g) (where k=1, 2, 3, . . . , 3g).

The above description relates to an LDPC-CC of a time varying period of3g and a coding rate of (n−1)/n (where n is an integer equal to orgreater than 2). Below, conditions are described for orders of anLDPC-CC of a time varying period of 3g and a coding rate of 1/2 (n=2).

Consider equations 15-1 to 15-3g as parity check polynomials of anLDPC-CC for which the time varying period is 3g (where g=1, 2, 3, 4, . .. ) and the coding rate is 1/2 (n=2).

[15]

(D ^(a#1,1,1) +D ^(a#1,1,2) +D ^(a#1,1,3))X(D)+(D ^(b#1,1) +D ^(b#1,2)+D ^(b#1,3))P(D)  (Equation 15-1)

(D ^(a#2,1,1) +D ^(a#2,1,2) +D ^(a#2,1,3))X(D)+(D ^(b#2,1) +D ^(b#2,2)+D ^(b#2,3))P(D)=0  (Equation 15-2)

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X(D)+(D ^(b#3,1) +D ^(b#3,2)+D ^(b#3,3))P(D)=0  (Equation 15-3)

(D ^(a#k,1,1) +D ^(a#k,1,2) +D ^(a#k,1,3))X(D)+(D ^(b#k,1) +D ^(b#k,2)+D ^(b#k,3))P(D)=0  (Equation 15-k)

(D ^(a#3g−2,1,1) +D ^(a#3g−2,1,2) +D ^(a#3g−2,1,3))X(D)+(D ^(b#3g−2,1)+D ^(b#3g−2,2) +D ^(b#3g−2,3))P(D)=0   (Equation 15-(3g−2))

(D ^(a#3g−1,1,1) +D ^(a#3g−1,1,2) +D ^(a#3g−1,1,3))X(D)+(D ^(b#3g−1,1)+D ^(b#3g−1,2) +D ^(b#3g−1,3))P(D)=0   (Equation 15-(3g−1))

(D ^(a#3g,1,1) +D ^(a#3g,1,2) +D ^(a#3g,1,3))X(D)+(D ^(b#3g,1) +D^(b#3g,2) +D ^(b#3g,3))P(D)=0  (Equation 15-3g)

At this time, X(D) is a polynomial representation of data (information)X and P(D) is a polynomial representation of parity. Here, in equations15-1 to 15-3g, parity check polynomials are assumed such that there arethree terms in X(D) and P(D) respectively.

Thinking in the same way as in the case of an LDPC-CC of a time varyingperiod of 3 and an LDPC-CC of a time varying period of 6, thepossibility of being able to obtain higher error correction capabilityis increased if the condition below (Condition #2-1>) is satisfied in anLDPC-CC of a time varying period of 3g and a coding rate of 1/2 (n=2)represented by parity check polynomials of equations 15-1 to 15-3g.

In an LDPC-CC of a time varying period of 3g and a coding rate of 1/2(n=2), parity and information at time i are represented by Pi andX_(1,1) respectively. If i %3g=k (where k=0, 1, 2, . . . , 3g−1) isassumed at this time, a parity check polynomial of equation 15-(k+1)holds true. For example, if i=2, i %3g=2 (k=2), and therefore equation16 holds true.

[16]

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(2,1)+(D ^(b#3,1) +D^(b#3,2) +D ^(b#3,3))P ₂=0  (Equation 16)

In equations 15-1 to 15-3g, it is assumed that a_(#k,1,1), a_(#k,1,2),and a_(#k,1,3) are integers (where a_(#k,1,1)≠a_(#k,1,2)≠a_(#k,1,3))(where k=1, 2, 3, . . . , 3g). Also, it is assumed that b_(#k,1),b_(#k,2), and b_(#k,3) are integers (where b_(#k,1)≠b_(#k,2)≠b_(#k,3)).A parity check polynomial of equation 15-k (k=1, 2, 3, . . . , 3g) iscalled “check equation #1c,” and a sub-matrix based on the parity checkpolynomial of equation 15-k is designated k-th sub-matrix H_(k). Next,an LDPC-CC of a time varying period of 3g is considered that isgenerated from first sub-matrix H₁, second sub-matrix H₂, thirdsub-matrix H₃, . . . , and 3g-th sub-matrix H_(3g)

<Condition #2-1>

In equations 15-1 to 15-3g, combinations of orders of X(D) and P(D)satisfy the following condition:

(a _(#1,1,1)%3,a _(#1,1,2)%3,a _(#1,1,3)%3) and (b _(#1,1)%3,b_(#1,2)%3,b _(#1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0);

(a _(#2,1,1)%3,a _(#2,1,2)%3,a _(#2,1,3)%3) and (b _(#2,1)%3,b_(#2,2)%3,b _(#2,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0);

(a _(#3,1,1)%3,a _(#3,1,2)%3,a _(#3,1,3)%3) and (b _(#3,1)%3,b_(#3,2)%3,b _(#3,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0);

(a _(#k,1,1)%3,a _(#k,1,2)%3,a _(#k,1,3)%3) and (b _(#10)%3,b_(#k,2)%3,b _(#k,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where k=1,2,3, . .. ,3g);

(a _(#3g−2,1,1)%3,a _(#3g−2,1,2)%3,a _(#3g−2,1,3)%3) and (b_(#3g−2,1)%3,b _(#3g−2,2)%3,b _(#3g−2,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0);

(a _(#3g−1,1,1)%3,a _(#3g−1,1,2)%3,a _(#3g−1,1,3)%3) and (b_(#3g−1,1)%3,b _(#3g−1,2)%3,b _(#3g−1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0); and

(a _(#3g,1,1)%3,a _(#3g,1,2)%3,a _(#3g,1,3)%3) and (b _(#3g,1)%3,b_(#3g,2)%3,b _(#3g,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0).

Taking ease of performing encoding into consideration, it is desirablefor one “0” to be present among the three items (b_(#k,1)%3, b_(#k,2)%3,b_(#k,3)%3) (where k=1, 2, . . . , 3g) in equations 15-1 to 15-3g. Thisis because of a feature that, if D⁰=1 holds true and b_(#k,1), b_(#k,2)and b_(#k,3) are integers equal to or greater than 0 at this time,parity P can be found sequentially.

Also, in order to provide relevancy between parity bits and data bits ofthe same point in time, and to facilitate a search for a code havinghigh correction capability, it is desirable for one “0” to be presentamong the three items (a_(#k,1,1)%3, a_(#k,1,2)%3, a_(#k,1,3)%3) (wherek=1, 2, . . . , 3g).

Next, an LDPC-CC of a time varying period of 3g (where g=2, 3, 4, 5, . .. ) that takes ease of encoding into account is considered. At thistime, if the coding rate is 1/2 (n=2), LDPC-CC parity check polynomialscan be represented as shown below.

[17]

(D ^(a#1,1,1) +D ^(a#1,1,2) +D ^(a#1,1,3))X(D)+(D ^(b#1,1) +D^(b#1,2)+1)P(D)  (Equation 17-1)

(D ^(a#2,1,1) +D ^(a#2,1,2) +D ^(a#2,1,3))X(D)+(D ^(b#2,1) +D^(b#2,2)+1)P(D)=0  (Equation 17-2)

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X(D)+(D ^(b#3,1) +D^(b#3,2)+1)P(D)=0  (Equation 17-3)

(D ^(a#k,1,1) +D ^(a#k,1,2) +D ^(a#k,1,3))X(D)+(D ^(b#k,1) +D^(b#k,2)+1)P(D)=0  (Equation 17-k)

(D ^(a#3g−2,1,1) +D ^(a#3g−2,1,2) +D ^(a#3g−2,1,3))X(D)+(D ^(b#3g−2,1)+D ^(b#3g−2,2)+1)P(D)=0  (Equation 17-(3g−2))

(D ^(a#3g−1,1,1) +D ^(a#3g−1,1,2) +D ^(a#3g−1,1,3))X(D)+(D ^(b#3g−1,1)+D ^(b#3g−1,2)+1)P(D)=0  (Equation 17-(3g−1))

(D ^(a#3g,1,1) +D ^(a#3g,1,2) +D ^(a#3g,1,3))X(D)+(D ^(b#3g,1) +D^(b#3g,2)+1)P(D)=0  (Equation 17-3g)

At this time, X(D) is a polynomial representation of data (information)X and P(D) is a polynomial representation of parity. Here, in equations17-1 to 17-3g, parity check polynomials are assumed such that there arethree terms in X(D) and P(D) respectively. In an LDPC-CC of a timevarying period of 3g and a coding rate of 1/2 (n=2), parity andinformation at time i are represented by Pi and X_(i,1) respectively. Ifi %3g=k (where k=0, 1, 2, . . . , 3g−1) is assumed at this time, aparity check polynomial of equation 17-(k+1) holds true. For example, ifi=2, i %3g=2 (k=2), and therefore equation 18 holds true.

[18]

(D ^(a#3,1,1) +D ^(a#3,1,2) +D ^(a#3,1,3))X _(2,1)+(D ^(b#3,1) +D^(b#3,2)+1)P ₂=0  (Equation 18)

If <Condition #3-1> and <Condition #4-1> are satisfied at this time, thepossibility of being able to create a code having higher errorcorrection capability is increased.

<Condition #3-1>

In equations 17-1 to 17-3g, combinations of orders of X(D) satisfy thefollowing condition:

(a _(#1,1,1)%3,a _(#1,1,2)%3,a _(#1,1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0);

(a _(#2,1,1)%3,a _(#2,1,2)%3,a _(#2,1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0);

(a _(#3,1,1)%3,a _(#3,1,2)%3,a _(#3,1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0);

(a _(#k,1,1)%3,a _(#k,1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0) (where k=1,2,3, . .. ,3g);

(a _(#3g−2,1,1)%3,a _(#3g−2,1,2)%3,a _(#3g−2,1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0);

(a _(#3g,1,1)%3,a _(#3g−1,1,2)%3,a _(#3g−1,1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0); and

(a _(#3g,1,1)%3,a _(#3g,1,2)%3,a _(#3g,1,3)%3) are any of(0,1,2),(0,2,1),(1,0,2),(1,2,0),(2,0,1), or (2,1,0).

In addition, in equations 17-1 to 17-3g, combinations of orders of P(D)satisfy the following condition:

(b _(#1,1)%3,b _(#1,2)%3),(b _(#2,1)%3,b _(#2,2)%3),(b _(#3,1)%3,b_(#3,2)%3), . . . ,(b _(#k,1)%3,b _(#k,2)%3),(b _(#3g−2,1)%3,b_(#3g−2,2)%3),(b _(#3g−1,1)%3,b _(#3g−1,2)%3), and (b _(#3g,1)%3,b_(#3g,2)%3) are any of (1,2), or (2,1) (k=1,2,3, . . . ,3g).

<Condition #3-1> has a similar relationship with respect to equations17-1 to 17-3g as <Condition #2-1> has with respect to equations 15-1 to15-3g. If the condition below (<Condition #4-1>) is added for equations17-1 to 17-3g in addition to <Condition #3-1>, the possibility of beingable to create an LDPC-CC having higher error correction capability isincreased.

<Condition #4-1>

Orders of P(D) of equations 17-1 to 17-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (b_(#1,1)%3g,b_(#1,2)%3g), (b_(#2,1)%3g, b_(#2,2)%3g), (b_(#3,1)%3g, b_(#3,2)%3g), .. . , (b_(k,1)%3g, b_(#k,2)%3g), . . . , (b_(#3g−2,1)%3g,b_(#3g−2,2)%3g), (b_(#3g−1,1)%3g, b_(#3g−1,2)%3g), and (b_(#3g,1)%3g,b_(#3g,2)%3g).

The possibility of obtaining good error correction capability is high ifthere is also randomness while regularity is maintained for positions atwhich “1”s are present in a parity check matrix. With an LDPC-CC forwhich the time varying period is 3g (where g=2, 3, 4, 5, . . . ) and thecoding rate is 1/2 (n=2) that has parity check polynomials of equations17-1 to 17-3g, if a code is created in which <Condition #4-1> is appliedin addition to <Condition #3-1>, it is possible to provide randomnesswhile maintaining regularity for positions at which “1”s are present ina parity check matrix, and therefore the possibility of obtaining bettererror correction capability is increased.

Next, an LDPC-CC of a time varying period of 3g (where g=2, 3, 4, 5, . .. ) is considered that enables encoding to be performed easily andprovides relevancy to parity bits and data bits of the same point intime. At this time, if the coding rate is 1/2 (n=2), LDPC-CC paritycheck polynomials can be represented as shown below.

[19]

(D ^(a#1,1,1) +D ^(a#1,1,2)+1)X(D)+(D ^(b#1,1) +D^(b#1,2)+1)P(D)=0  (Equation 19-1)

(D ^(a#2,1,1) +D ^(a#2,1,2)+1)X(D)+(D ^(b#2,1) +D^(b#2,2)+1)P(D)=0  (Equation 19-2)

(D ^(a#3,1,1) +D ^(a#3,1,2)+1)X(D)+(D ^(b#3,1) +D^(b#3,2)+1)P(D)=0  (Equation 19-3)

(D ^(a#k,1,1) +D ^(a#k,1,2)+1)X(D)+(D ^(b#k,1) +D^(b#k,2)+1)P(D)=0  (Equation 19-k)

(D ^(a#3g−2,1,1) +D ^(a#3g−2,1,2)+1)X(D)+(D ^(b#3g−2,1) +D^(b#3g−2,2)+1)P(D)=0  (Equation 19-(3g−2))

(D ^(a#3g−1,1,1) +D ^(a#3g−1,1,2)+1)X(D)+(D ^(b#3g−1,1) +D^(b#3g−1,2)+1)P(D)=0  (Equation 19-(3g−1))

(D ^(a#3g,1,1) +D ^(a#3g,1,2)+1)X(D)+(D ^(b#3g,1) +D^(b#3,2)+1)P(D)=0  (Equation 19-3g)

At this time, X(D) is a polynomial representation of data (information)X and P(D) is a polynomial representation of parity.

In equations 19-1 to 19-3g, parity check polynomials are assumed suchthat there are three terms in X(D) and P(D) respectively, and a D⁰ termis present in X(D) and P(D) (where k=1, 2, 3, . . . , 3g).

In an LDPC-CC of a time varying period of 3g and a coding rate of 1/2(n=2), parity and information at time i are represented by Pi andX_(1,1) respectively. If i %3g=k (where k=0, 1, 2, . . . , 3g−1) isassumed at this time, a parity check polynomial of equation 19-(k+1)holds true. For example, if i=2, i %3g=2 (k=2), and therefore equation20 holds true.

[20]

(D ^(a#3,1,1) +D ^(a#3,1,2)+1)X _(2,1)+(D ^(b#3,1) +D ^(b#3,2)+1)P₂(D)=0  (Equation 20)

If following <Condition #5-1> and <Condition #6-1> are satisfied at thistime, the possibility of being able to create a code having higher errorcorrection capability is increased.

<Condition #5-1>

In equations 19-1 to 19-3g, combinations of orders of X(D) satisfy thefollowing condition:

(a _(#1,1,1)%3,a _(#1,1,2)%3) is (1,2) or (2,1);

(a _(#2,1,1)%3,a _(#2,1,2)%3) is (1,2) or (2,1);

(a _(#3,1,1)%3,a _(#3,1,2)%3) is (1,2) or (2,1);

(a _(#k,1,1)%3,a _(#k,1,2)%3) is (1,2) or (2,1) (where k=1,2,3, . . .,3g);

(a _(#3g−2,1,1)%3,a _(#3g−2,1,2)%3) is (1,2) or (2,1),

(a _(#3g−1,1,1)%3,a _(#3g−1,1,2)%3) is (1,2) or (2,1); and

(a _(#3g,1,1)%3,a _(#3g,1,2)%3) is (1,2) or (2,1).

In addition, in equations 19-1 to 19-3g, combinations of orders of P(D)satisfy the following condition: (b_(#1,1)%3, b_(#1,2)%3), (b_(#2,1)%3,b_(#2,2)%3), (b_(#3,1)%3, b_(#3,2)%3), (b_(#10)%3, b_(#k,2)%3),(b_(#3g−2,1)%3, b_(#3g−2,2)%3), (b_(#3g−1,1)%3, b_(#3g−1,2)%3), and(b_(#3g,1)%3, b_(#3g,2)%3) are any of (1, 2), or (2, 1) (where k=1, 2,3, . . . , 3g).

<Condition #5-1> has a similar relationship with respect to equations19-1 to 19-3g as <Condition #2-1> has with respect to equations 15-1 to15-3g. If the condition below (<Condition #6-1>) is added for equations19-1 to 19-3g in addition to <Condition #5-1>, the possibility of beingable to create an LDPC-CC having higher error correction capability isincreased.

<Condition #6-1>

Orders of X(D) of equations 19-1 to 19-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,1,1)%3g,a_(#1,1,2)%3g), (a_(#2,1,1)%3g, a_(#2,1,2)%3g), . . . , (a_(#p,1,1)%3g,a_(#p,1,2)%3g), . . . , (a_(#3g,1,1)%3g, a_(#3g,1,2)%3g) (where p=1, 2,3, . . . , 3g); and

Orders of P(D) of equations 19-1 to 19-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (b_(#1,1)%3g,b_(#1,2)%3g), (b_(#2,1)%3g, b_(#2,2)%3g), (b_(#3,1)%3g, b_(#3,2)%3g), .. . , (b_(k,1)%3g, b_(#k,2)%3g), . . . , (b_(#3g−2,1)%3g,b_(#3g−2,2)%3g), (b_(#3g−1,1)%3g, b_(#3g−1,2)%3g), and (b_(#3g,1)%3g,b_(#3g,2)%3g) (where k=1, 2, 3, . . . 3g).

The possibility of obtaining good error correction capability is high ifthere is also randomness while regularity is maintained for positions atwhich “1”s are present in a parity check matrix. With an LDPC-CC forwhich the time varying period is 3g (where g=2, 3, 4, 5, . . . ) and thecoding rate is 1/2 that has parity check polynomials of equations 19-1to 19-3g, if a code is created in which <Condition #6-1> is applied inaddition to <Condition #5-1>, it is possible to provide randomness whilemaintaining regularity for positions at which “1”s are present in aparity check matrix, so that the possibility of obtaining better errorcorrection capability is increased.

The possibility of being able to create a code having higher errorcorrection capability is also increased if a code is created using<Condition #6′-1> instead of <Condition #6-1>, that is, using <Condition#6′-1> in addition to <Condition #5-1>.

<Condition #6′-1>

Orders of X(D) of equations 19-1 to 19-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (a_(#1,1,1)%3g,a_(#1,1,2)%3g), (a_(#2,1,1)%3g, a_(#2,1,2)%3g), . . . , (a_(#0,1)%3g,a_(#p,1,2)%3g), . . . , and (a_(#3g,1,1)%3g, a_(#3g,1,2)%3g) (where p=1,2, 3, . . . , 3g); or

Orders of P(D) of equations 19-1 to 19-3g satisfy the followingcondition: all values other than multiples of 3 (that is, 0, 3, 6, . . ., 3g−3) from among integers from 0 to 3g−1 (0, 1, 2, 3, 4, . . . , 3g−2,3g−1) are present in the following 6g values of (b_(#1,1)%3g,b_(#1,2)%3g), (b_(#2,1)%3g, b_(#2,2)%3g), (b_(#3,1)%3g, b_(#3,2)%3g), .. . , (b_(#k,1)%3g, b_(#k,2)%3g, . . . , (b_(#3g−2,1)%3g,b_(#3g−2,2)%3g), (b_(#3g−1,1)%3g, b_(#3g−1,2)%3g) and (b_(#3g,1)%3g,b_(#3g,2)%3g) (where k=1, 2, 3, . . . , 3g).

Examples of LDPC-CCs of a coding rate of 1/2 and a time varying periodof 6 having good error correction capability are shown in Table 4.

TABLE 4 Code Parity check polynomial LDPC-CC #1 Check polynomial #1:(D³²⁸ + D³¹⁷ + 1)X(D) + (D⁵⁸⁹ + D⁴³⁴ + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁵⁹⁶ + D⁵⁵³ + 1)X(D) + (D⁵⁸⁶ + D⁴⁶¹ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁵⁵⁰ + D¹⁴³ + 1)X(D) + (D⁴⁷⁰ + D⁴⁴⁸ +1)P(D) = 0 of 6 and a Check polynomial #4: (D⁴⁷⁰ + D²²³ + 1)X(D) +(D²⁵⁶ + D⁴¹ + 1)P(D) = 0 coding rate of Check polynomial #5: (D⁸⁹ +D⁴⁰ + 1)X(D) + (D³¹⁶ + D⁷¹ + 1)P(D) = 0 1/2 Check polynomial #6: (D³²⁰ +D¹⁹⁰ + 1)X(D) + (D⁵⁷⁵ + D¹³⁶ + 1)P(D) = 0 LDPC-CC #2 Check polynomial#1: (D⁵²⁴ + D⁵¹¹ + 1)X(D) + (D²¹⁵ + D¹⁰³ + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁵⁴⁷ + D²⁸⁷ + 1)X(D) + (D⁴⁶⁷ + D¹ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D²⁸⁹ + D⁶² + 1)X(D) + (D⁵⁰³ + D⁵⁰² + 1)P(D)= 0 of 6 and a Check polynomial #4: (D⁴⁰¹ + D⁵⁵ + 1)X(D) + (D⁴⁴³ +D¹⁰⁶ + 1)P(D) = 0 coding rate of Check polynomial #5: (D⁴³³ + D³⁹⁵ +1)X(D) + (D⁴⁰⁴ + D¹⁰⁰ + 1)P(D) = 0 1/2 Check polynomial #6: (D¹³⁶ +D⁵⁹ + 1)X(D) + (D⁵⁹⁹ + D⁵⁵⁹ + 1)P(D) = 0 LDPC-CC #3 Check polynomial #1:(D²⁵³ + D⁴⁴ + 1)X(D) + (D⁴⁷³ + D²⁵⁶ + 1)P(D) = 0 of a time Checkpolynomial #2: (D⁵⁹⁵ + D¹⁴³ + 1)X(D) + (D⁵⁹⁸ + D⁹⁵ + 1)P(D) = 0 varyingperiod Check polynomial #3: (D⁹⁷ + D¹¹ + 1)X(D) + (D⁵⁹² + D⁴⁹¹ + 1)P(D)= 0 of 6 and a Check polynomial #4: (D⁵⁰ + D¹⁰ + 1)X(D) + (D³⁶⁸ + D¹¹² +1)P(D) = 0 coding rate of Check polynomial #5: (D²⁸⁶ + D²²¹ + 1)X(D) +(D⁵¹⁷ + D³⁵⁹ + 1)P(D) = 0 1/2 Check polynomial #6: (D⁴⁰⁷ + D³²² +1)X(D) + (D²⁸³ + D²⁵⁷ + 1)P(D) = 0

An LDPC-CC of a time varying period of g with good characteristics hasbeen described above. Also, for an LDPC-CC, it is possible to provideencoded data (codeword) by multiplying information vector n by generatormatrix G. That is, encoded data (codeword) c can be represented byc=n×G. Here, generator matrix G is found based on parity check matrix Hdesigned in advance. To be more specific, generator matrix G refers to amatrix satisfying G×H^(T)=0.

For example, a convolutional code of a coding rate of 1/2 and generatorpolynomial G=[1 G₁(D)/G₀(D)] will be considered as an example. At thistime, G₁ represents a feed-forward polynomial and G₀ represents afeedback polynomial. If a polynomial representation of an informationsequence (data) is X(D), and a polynomial representation of a paritysequence is P(D), a parity check polynomial is represented as shown inequation 21 below.

[21]

G ₁(D)X(D)+G ₀(D)P(D)=0   (Equation 21)

where D is a delay operator.

FIG. 5 shows information relating to a (7, 5) convolutional code. A (7,5) convolutional code generator polynomial is represented as G=[1(D²+1)/(D²+D+1)]. Therefore, a parity check polynomial is as shown inequation 22 below.

[22]

(D ²+1)X(D)+(D ² +D+1)P(D)=0  (Equation 22)

Here, data at point in time i is represented by X_(i), and parity byP_(i), and transmission sequence Wi is represented as W_(i)=(X_(i),P_(i)). Then transmission vector w is represented as w=(X₁, P₁, X₂, P₂,. . . , X_(i), P_(i) . . . )^(T). Thus, from equation 22, parity checkmatrix H can be represented as shown in FIG. 5. At this time, therelational equation in equation 23 below holds true.

[23]

Hw=0  (Equation 23)

Therefore, with parity check matrix H, the decoding side can performdecoding using belief propagation (BP) decoding, min-sum decodingsimilar to BP decoding, offset BP decoding, normalized BP decoding,shuffled BP decoding, or suchlike belief propagation, as shown inNon-Patent Literature 5, Non-Patent Literature 6 and Non-PatentLiterature 9.

(Time-invariant/time varying LDPC-CCs (of a coding rate of (n−1)/n)based on a convolutional code (where n is a natural number))

An overview of time-invariant/time varying LDPC-CCs based on aconvolutional code is given below.

A parity check polynomial represented as shown in equation 24 will beconsidered, with polynomial representations of coding rate of R=(n−1)/nas information X₁, X₂, . . . , X_(n−1) as X₁(D), X₂(D), . . . ,X_(n−1)(D), and a polynomial representation of parity P as P(D).

[24]

(D ^(a) _(1,1) +D ^(a) _(1,2) + . . . +D ^(a) _(1,r1)+1)X ₁(D)+(D ^(a)_(2,1) +D ^(a) _(2,2) + . . . +D ^(a) _(2,r2)+1)X ₂(D)+ . . . +(D ^(a)_(n−)1,1+D ^(a) _(n−)1,2+ . . . +D ^(a) _(n−1,r) _(n−) 1+1)X_(n−1)(D)+(D ^(b) ₁ +D ^(b) ₂+ . . . +^(b) _(s)+1)P(D)=0  (Equation 24)

In equation 24, at this time, a_(p,p) (where p=1, 2, . . . , n−1 andq=1, 2, . . . , rp) is, for example, a natural number, and satisfies thecondition a_(p,1)≠a_(p,2)≠ . . . ≠a_(p,rp). Also, b_(q) (where q=1, 2, .. . , s) is a natural number, and satisfies the condition b₁≠b₂≠ . . .≠b_(s). A code defined by a parity check matrix based on a parity checkpolynomial of equation 24 at this time is called a time-invariantLDPC-CC here.

Here, m different parity check polynomials based on equation 24 areprovided (where m is an integer equal to or greater than 2). Theseparity check polynomials are represented as shown below.

[25]

A _(X1,i)(D)X ₁(D)+A _(X2,i)(D)X ₂(D)+ . . . +A _(Xn−1,i)(D)X_(n−1)(D)+B _(i)(D)P(D)=0  (Equation 25)

Here, i=0, 1, . . . , m−1.

Then information X₁, X₂, . . . , X_(n−1) at point in time j isrepresented as X_(1,j), X_(2,j), . . . , X_(n−1,j), parity P at point intime j is represented as P_(j), and u_(j)=(X_(1,j), X_(2,j), . . . ,X_(n−1,j), P_(j))^(T). At this time, information X_(1,j), X_(2,j), . . ., X_(n−1,j), and parity P_(j) at point in time j satisfy a parity checkpolynomial of equation 26.

[26]

A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+ . . . +A _(Xn−1,k)(D)X_(n−1)(D)+B _(k)(D)P(D)=0  (Equation 26)

Here, “j mod m” is a remainder after dividing j by m.

A code defined by a parity check matrix based on a parity checkpolynomial of equation 26 is called a time varying LDPC-CC here. At thistime, a time-invariant LDPC-CC defined by a parity check polynomial ofequation 24 and a time varying LDPC-CC defined by a parity checkpolynomial of equation 26 have a characteristic of enabling parityeasily to be found sequentially by means of a register and exclusive OR.

For example, the configuration of parity check matrix H of an LDPC-CC ofa time varying period of 2 and a coding rate of 2/3 based on equation 24to equation 26 is shown in FIG. 6. Two different check polynomials of atime varying period of 2 based on equation 26 are designed “checkequation #1” and “check equation #2.” In FIG. 6, (Ha,111) is a partcorresponding to “check equation #1,” and (Hc,111) is a partcorresponding to “check equation #2.” Below, (Ha,111) and (Hc,111) aredefined as sub-matrices.

Thus, LDPC-CC parity check matrix H of a time varying period of 2 ofthis proposal can be defined by a first sub-matrix representing a paritycheck polynomial of “check equation #1”, and by a second sub-matrixrepresenting a parity check polynomial of “check equation #2”.Specifically, in parity check matrix H, a first sub-matrix and secondsub-matrix are arranged alternately in the row direction. When thecoding rate is 2/3, a configuration is employed in which a sub-matrix isshifted three columns to the right between an i'th row and (i+1)-th row,as shown in FIG. 6.

In the case of a time varying LDPC-CC of a time varying period of 2, ani'th row sub-matrix and an (i+1)-th row sub-matrix are differentsub-matrices. That is to say, either sub-matrix (Ha,111) or sub-matrix(Hc,111) is a first sub-matrix, and the other is a second sub-matrix. Iftransmission vector u is represented as u=(X_(1,0), X_(2,0), P₀,X_(1,1), X_(2,1), P₁, . . . , X_(1,k), X_(2,k), P_(k), . . . )^(T), therelationship Hu=0 holds true (see equation 23).

Next, an LDPC-CC for which the time varying period is m is considered inthe case of a coding rate of 2/3. In the same way as when the timevarying period is 2, m parity check polynomials represented by equation24 are provided. Then “check equation #1” represented by equation 24 isprovided. “Check equation #2” to “check equation #m” represented byequation 24 are provided in a similar way. Data X and parity P of pointin time mi+1 are represented by X_(mi+1) and P₊₁ respectively, data Xand parity P of point in time mi+2 are represented by X_(mi+2) and P₊₂respectively, . . . , and data X and parity P of point in time mi+m arerepresented by X_(mi+m) and P_(mi+m) respectively (where i is aninteger).

Consider an LDPC-CC for which parity P_(mi+1) of point in time mi+1 isfound using “check equation #1,” parity P_(mi+2) of point in time mi+2is found using “check equation #2,” . . . , and parity P_(mi+m) of pointin time mi+m is found using “check equation #m.” An LDPC-CC code of thiskind provides the following advantages:

-   -   An encoder can be configured easily, and parity can be found        sequentially.    -   Termination bit reduction and received quality improvement in        puncturing upon termination can be expected.

FIG. 7 shows the configuration of the above LDPC-CC parity check matrixof a coding rate of 2/3 and a time varying period of m. In FIG. 7,(H₁,111) is a part corresponding to “check equation #1,” (H₂,111) is apart corresponding to “check equation #2,” . . . , and (H_(in),111) is apart corresponding to “check equation #m.” Below, (E1₁,111) is definedas a first sub-matrix, (H₂,111) is defined as a second sub-matrix, . . ., and (H_(m), 111) is defined as an m-th sub-matrix.

Thus, LDPC-CC parity check matrix H of a time varying period of m ofthis proposal can be defined by a first sub-matrix representing a paritycheck polynomial of “check equation #1”, a second sub-matrixrepresenting a parity check polynomial of “check equation #2”, . . . ,and an m-th sub-matrix representing a parity check polynomial of “checkequation #m”. Specifically, in parity check matrix H, a first sub-matrixto m-th sub-matrix are arranged periodically in the row direction (seeFIG. 7). When the coding rate is 2/3, a configuration is employed inwhich a sub-matrix is shifted three columns to the right between an i-throw and (i+1)-th row (see FIG. 7).

If transmission vector u is represented as u=(X_(1,0), X_(2,0), P₀,X_(1,1), X_(2,1), P₁, . . . , X_(1,k), X_(2,k), P_(k), . . . )^(T), therelationship Hu=0 holds true (see equation 23).

In the above description, a case of a coding rate of 2/3 has beendescribed as an example of a time-invariant/time varying LDPC-CC basedon a convolutional code of a coding rate of (n−1)/n, but atime-invariant/time varying LDPC-CC parity check matrix based on aconvolutional code of a coding rate of (n−1)/n can be created bythinking in a similar way.

That is to say, in the case of a coding rate of 2/3, in FIG. 7, (H₁,111)is a part (first sub-matrix) corresponding to “check equation #1,”(H₂,111) is a part (second sub-matrix) corresponding to “check equation#2,” . . . , and (H_(m),111) is a part (m-th sub-matrix) correspondingto “check equation #m,” while, in the case of a coding rate of (n−1)/n,the situation is as shown in FIG. 8. That is to say, a part (firstsub-matrix) corresponding to “check equation #1” is represented by(H₁,11 . . . 1), and a part (k-th sub-matrix) corresponding to “checkequation #k” (where k=2, 3, . . . , m) is represented by (H_(k),11 . . .1). At this time, the number of “1”s of parts excluding H_(k) in thek-th sub-matrix is n−1. Also, in parity check matrix H, a configurationis employed in which a sub-matrix is shifted n−1 columns to the rightbetween an i'th row and (i+1)-th row (see FIG. 8).

If transmission vector u is represented as u=(X_(1,0), X_(2,0), P₀,X_(1,1), X_(2,1), . . . , X_(n−1,1), P₁, . . . , X_(1,k), X_(2,k), . . ., X_(n−1,k), P_(k), . . . )^(T), the relationship Hu=0 holds true (seeequation 23).

FIG. 9 shows an example of the configuration of an LDPC-CC encoder whenthe coding rate is R=1/2.

As shown in FIG. 9, LDPC-CC encoder 100 is provided mainly with datacomputing section 110, parity computing section 120, weight controlsection 130, and modulo 2 adder (exclusive OR computer) 140.

Data computing section 110 is provided with shift registers 111-1 to111-M and weight multipliers 112-0 to 112-M.

Parity computing section 120 is provided with shift registers 121-1 to121-M and weight multipliers 122-0 to 122-M.

Shift registers 111-1 to 111-M and 121-1 to 121-M are registers storingand v_(2,t−i) (where i=0, M) respectively, and, at a timing at which thenext input comes in, send a stored value to the adjacent shift registerto the right, and store a new value sent from the adjacent shiftregister to the left. The initial state of the shift registers isall-zeros.

Weight multipliers 112-0 to 112-M and 122-0 to 122-M switch values of h₁^((m)) and h₂ ^((m)) to 0 or 1 in accordance with a control signaloutputted from weight control section 130.

Based on a parity check matrix stored internally, weight control section130 outputs values of h₁ ^((m)) and h₂ ^((m)) at that timing, andsupplies them to weight multipliers 112-0 to 112-M and 122-0 to 122-M.

Modulo 2 adder 140 adds all modulo 2 calculation results to the outputsof weight multipliers 112-0 to 112-M and 122-0 to 122-M, and calculatesv_(2,t).

By employing this kind of configuration, LDPC-CC encoder 100 can performLDPC-CC encoding in accordance with a parity check matrix.

If the arrangement of rows of a parity check matrix stored by weightcontrol section 130 differs on a row-by-row basis, LDPC-CC encoder 100is a time varying convolutional encoder. Also, in the case of an LDPC-CCof a coding rate of (q−1)/q, a configuration needs to be employed inwhich (q−1) data computing sections 110 are provided and modulo 2 adder140 performs modulo 2 addition (exclusive OR computation) of the outputsof weight multipliers.

Embodiment 1

Next, the present embodiment will explain a search method that cansupport a plurality of coding rates in a low computational complexity inan encoder and decoder. By using an LDPC-CC searched out by the methoddescribed below, it is possible to realize high data received quality inthe decoder.

With the LDPC-CC search method according to the present embodiment,LDPC-CCs of coding rates of 2/3, 3/4, 4/5, . . . , (q−1)/q aresequentially searched based on, for example, a coding rate of 1/2 amongLDPC-CCs with good characteristics described above. By this means, incoding and decoding processing, by preparing a coder and decoder in thehighest coding rate of (q−1)/q, it is possible to perform coding anddecoding in a coding rate of (s−1)/s (S=2, 3, . . . , q−1) lower thanthe highest coding rate of (q−1)/q.

A case in which the time varying period is 3 will be described below asan example. As described above, an LDPC-CC for which the time varyingperiod is 3 can provide excellent error correction capability.

(LDPC-CC search method)

(1) Coding rate of 1/2

First, an LDPC-CC of a coding rate of 1/2 is selected as a referenceLDPC-CC of a coding rate of 1/2. Here, an LDPC-CC of goodcharacteristics described above is selected as a reference LDPC-CC of acoding rate of 1/2.

A case will be explained below where the parity check polynomialsrepresented by equations 27-1 to 27-3 are used as parity checkpolynomials of a reference LDPC-CC of a coding rate of 1/2. The examplesof equations 27-1 to 27-3 are represented in the same way as above (i.e.an LDPC-CC of good characteristics), so that it is possible to define anLDPC-CC of a time varying period of 3 by three parity check polynomials.

[27]

(D ³⁷³ +D ⁵⁶+1)X ₁(D)+(D ⁴⁰⁶ +D ²¹⁸+1)P(D)=0  (Equation 27-1)

(D ⁴⁵⁷ +D ¹⁹⁷+1)X ₁(D)+(D ⁴⁹¹ +D ²²+1)P(D)=0  (Equation 27-2)

(D ⁴⁸⁵ +D ⁷⁰+1)X ₁(D)+(D ²³⁶ +D ¹⁸¹+1)P(D)=0  (Equation 27-1)

As described in table 3, equations 27-1 to 27-3 are represented as anexample of an LDPC-CC with good characteristics where the time varyingperiod is 3 and the coding rate is 1/2. Then, as described above (withan LDPC-CC of good characteristics), information X₁ at point in time jis represented as X_(1,j), parity P at point in time j is represented asP_(j), and u_(j)=(X_(1,j), P_(j))^(T). At this time, information X_(1,j)and parity P_(j) at point in time j satisfy a parity check polynomial ofequation 27-1 when “j mod 3=0.” Further, information X_(1,j) and parityP_(j) at point in time j satisfy a parity check polynomial of equation27-2 when “j mod 3=1.” Further, information X_(1,j) and parity P_(j) atpoint in time j satisfy a parity check polynomial of equation 27-3 when“j mod 3=2.” At this time, the relationships between parity checkpolynomials and a parity check matrix are the same as above (i.e. as inan LDPC-CC of good characteristics).

(2) Coding rate of 2/3

Next, LDPC-CC parity check polynomials of a coding rate of 2/3 iscreated based on the parity check polynomials of a coding rate of 1/2with good characteristics. To be more specific, LDPC-CC parity checkpolynomials of a coding rate of 2/3 are formed, including the referenceparity check polynomials of a coding rate of 1/2.

As shown in equations 28-1 to 28-3, upon using equations 27-1 to 27-3 ina reference LDPC-CC of a coding rate of 1/2, it is possible to representLDPC-CC parity check polynomials of a coding rate of 2/3.

[28]

(D ³⁷³ +D ⁵⁶+1)X ₁(D)+(D ^(α1) +D ^(β1)+1)X ₂(D)+D ⁴⁰⁶ +D²¹⁸+1)P(D)=0  (Equation 28-1)

(D ⁴⁵⁷ +D ¹⁹⁷+1)X ₁(D)+(D ^(α2) +D ^(β2)+1)X ₂(D)+D ⁴⁹¹ +D²²+1)P(D)=0  (Equation 28-1)

(D ⁴⁸⁵ +D ⁷⁰+1)X ₁(D)+(D ^(α3) +D ^(β3)+1)X ₂(D)+D ²³⁶ +D¹⁸¹+1)P(D)=0  (Equation 28-1)

The parity check polynomials represented by equations 28-1 to 28-3 areformed by adding term X2(D) to equations 27-1 to 27-3. LDPC-CC paritycheck polynomials of a coding rate of 2/3 used in equations 28-1 to 28-3are references for parity check polynomials of a coding rate of 3/4.

Also, in equations 28-1 to 28-3, if the orders of X2(D), (α1, β1), (α2,β2), (α3, β3), are set to satisfy the above conditions (e.g. <Condition#1> to <Condition #6>), it is possible to provide an LDPC-CC of goodcharacteristics even in a coding rate of 2/3.

Then, as described above (with an LDPC-CC of good characteristics),information X₁ and X₂ at point in time j is represented as X_(1,j) andX_(2,j), parity P at point in time j is represented as P_(j), andu_(j)=(X_(1,j), X_(2,j), P_(j))^(T). At this time, information X_(1,j)and X_(2,j) and parity P_(j) at point in time j satisfy a parity checkpolynomial of equation 28-1 when “j mod 3=0.” Further, informationX_(1,j) and X_(2,j) and parity P_(j) at point in time j satisfy a paritycheck polynomial of equation 28-2 when “j mod 3=1.” Further, informationX_(1,j) and X_(2,j) and parity P_(j) at point in time j satisfy a paritycheck polynomial of equation 28-3 when “j mod 3=2.” At this time, therelationships between parity check polynomials and a parity check matrixare the same as above (i.e. as in an LDPC-CC of good characteristics).

(3) Coding rate of 3/4

Next, LDPC-CC parity check polynomials of a coding rate of 3/4 iscreated based on the above parity check polynomials of a coding rate of2/3. To be more specific, LDPC-CC parity check polynomials of a codingrate of 3/4 are formed, including the reference parity check polynomialsof a coding rate of 2/3.

Equations 29-1 to 29-3 show LDPC-CC parity check polynomials of a codingrate of 3/4 upon using equations 28-1 to 28-3 in a reference LDPC-CC ofa coding rate of 2/3.

[29]

(D ³⁷³ +D ⁵⁶+1)X ₁(D)+(D ^(α1) +D ^(β1)+1)X ₂(D)+(D ^(γ1) +D ^(δ1)+1)X₃(D)+(D ⁴⁰⁶ +D ²¹⁸+1)P(D)=0  (Equation 29-1)

(D ⁴⁵⁷ +D ¹⁹⁷+1)X ₁(D)+(D ^(α2) +D ^(β2)+1)X ₂(D)+(D ^(γ2) +D ^(δ2)+1)X₃(D)+(D ⁴⁹¹ +D ²²+1)P(D)=0  (Equation 29-2)

(D ⁴⁸⁵ +D ⁷⁰+1)X ₁(D)+(D ^(α3) +D ^(β3)+1)X ₂(D)+(D ^(γ3) +D ^(δ3)+1)X₃(D)+(D ²³⁶ +D ¹⁸¹+1)P(D)=0  (Equation 29-3)

The parity check polynomials represented by equations 29-1 to 29-3 areformed by adding term X2(D) to equations 28-1 to 28-3. Also, inequations 29-1 to 29-3, if the orders in X2(D), (γ1, δ1), (γ2, δ2), (γ3,δ3), are set to satisfy the above conditions (e.g. <Condition #1> to<Condition #6>) with good characteristics, it is possible to provide anLDPC-CC of good characteristics even in a coding rate of 3/4.

Then, as described above (LDPC-CC of good characteristics), informationX₁, X₂ and X₃ at point in time j is represented as X_(1,j), X_(2,j) andX_(3,j), parity P at point in time j is represented as P_(j), andu_(j)=(X_(1,j), X₂, P_(j))^(T). At this time, information X₁, X_(2,j)and X_(3,j) and parity P_(j) at point in time j satisfy a parity checkpolynomial of equation 29-1 when “j mod 3=0.” Further, informationX_(1,j), X_(2,j) and X_(3,j) and parity P_(j) at point in time j satisfya parity check polynomial of equation 29-2 when “j mod 3=1.” Further,information X_(i,j), X_(2,j) and X_(3j) and parity P_(j) at point intime j satisfy a parity check polynomial of equation 29-3 when “j mod3=2.” At this time, the relationships between parity check polynomialsand a parity check matrix are the same as above (i.e. as in an LDPC-CCof good characteristics).

Equations 30-1 to 30-(q−1) show general LDPC-CC parity check polynomialsof a time varying period of g upon performing a search as above.

[30]

A _(X1,k)(D)X ₁(D)+B _(k)(D)P(D)=0(k=i mod g)  (Equation 30-1)

A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+B _(k)(D)P(D)=0(k=i modg)  (Equation 30-2)

A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+A _(X3,k)(D)X ₃(D)+B_(k)(D)P(D)=0(k=i mod g)  (Equation 30-3)

A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+ . . . A _(Xq−1,k)(D)X _(q−1)(D)=B_(k)(D)P(D)=0(k=i mod g)  (Equation 30-(q−1))

Here, equation 30-1 is represented as above because it is a generalequation. However, as described above (with an LDPC-CC of goodcharacteristics), the time varying period is g, and therefore equation30-1 is actually represented by g parity check polynomials. For example,as described with the present embodiment, when the time varying periodis 3, representation of three parity check polynomials is provided asshown in equations 27-1 to 27-3. Similar to equation 30-1, equations30-2 to 30-(q−1) each have a time varying period of g, and therefore arerepresented by g parity check equations.

Here, assume that g parity check equations of equation 30-1 arerepresented by equation 30-1-0, equation 30-1-1, equation 30-1-2, . . ., equation 30-1-(g−2) and equation 30-1-(g−1).

Similarly, equation 30-w is represented by g parity check polynomials(w=2, 3, . . . , q−1). Here, assume that g parity check equations ofequation 30-w are represented by equation 30-w-0, equation 30-w-1,equation 30-w-2, . . . , equation 30-w-(g−2) and equation 30-w-(g−1).

Also, in equations 30-1 to 30-(q−1), information X₁, X₂, . . . , X_(q−1)at point in time i is represented as X_(1,i), X_(2,i), . . . ,X_(q−1,i), and parity P at point in time i is represented as P₁. Also,A_(Xr,k)(D) refers to a term of X_(r)(D) in the parity check polynomialfor k calculated from “k=i mod g,” at point in time i where the codingrate is (r−1)/r (r=2, 3, . . . , q, and q is a natural number equal toor greater than 3). Also, B_(k)(D) refers to a term of P(D) in theparity check polynomial for k calculated from “k=i mod g,” at point intime i where the coding rate is (r−1)/r. Here, “i mod g” is a remainderafter dividing i by g.

That is, equation 30-1 represents an LDPC-CC parity check polynomial ofa time varying period of g supporting a coding rate of 1/2, equation30-2 represents an LDPC-CC parity check polynomial of a time varyingperiod of g supporting a coding rate of 2/3, . . . , and equation30-(q−1) represents an LDPC-CC parity check polynomial of a time varyingperiod of g supporting a coding rate of (q−1)/q.

Thus, based on equation 30-1 which represents an LDPC-CC parity checkpolynomial of a coding rate of 1/2 with good characteristics, an LDPC-CCparity check polynomial of a coding rate of 2/3 (i.e. equation 30-2) isgenerated.

Further, based on equation 30-2 which represents an LDPC-CC parity checkpolynomial of a coding rate of 2/3, an LDPC-CC parity check polynomialof a coding rate of 3/4 (i.e. equation 30-3) is generated. The sameapplies to the following, and, based on an LDPC-CC of a coding rate of(r−1)/r, LDPC-CC parity check polynomials of a coding rate of r/(r+1)(r=2, 3, . . . , q−2, q−1) are generated.

The above method of forming parity check polynomials will be shown in adifferent way. Consider an LDPC-CC for which the coding rate is (y−1)/yand the time varying period is g, and an LDPC-CC for which the codingrate is (z−1)/z and the time varying period is g. Here, the maximumcoding rate is (q−1)/q among coding rates to share encoder circuits andto share decoder circuits, where g is an integer equal to or greaterthan 2, y is an integer equal to or greater than 2, z is an integerequal to or greater than 2, and the relationship of y<z≤q holds true.Here, sharing encoder circuits means to share circuits inside encoders,and does not mean to share circuits between an encoder and a decoder.

At this time, if w=y−1 is assumed in equations 30-w-0, 30-w-1, 30-w-2, .. . , 30-w-(g−2) and 30-w-(g−1), which represent g parity checkpolynomials described upon explaining equations 30-1 to 30-(q−1),representations of g parity check polynomials is provided as shown inequations 31-1 to 31-g.

$\begin{matrix}{\mspace{79mu} \lbrack 31\rbrack} & \; \\{{{{A_{{X\; 1},0}(D)}{X_{1}(D)}} + {{A_{{X\; 2},0}(D)}{X_{2}(D)}} + \ldots + {{A_{{{Xy} - 1},0}(D)}{X_{y - 1}(D)}} + {{B_{0}(D)}{P(D)}}} = {0\mspace{14mu} ( {0 = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}1} ) \\{{{{B_{0}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},0}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {0 = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}1^{\prime}} ) \\{{{{A_{{X\; 1},1}(D)}{X_{1}(D)}} + {{A_{{X\; 2},1}(D)}{X_{2}(D)}} + \ldots + {{A_{{{Xy} - 1},1}(D)}{X_{y - 1}(D)}} + {{B_{1}(D)}{P(D)}}} = {0\mspace{14mu} ( {1 = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}2} ) \\\begin{matrix}{{{{B_{1}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},1}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {1 = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} \\{\mspace{79mu} \vdots}\end{matrix} & ( {{Equation}\mspace{14mu} 31\text{-}2^{\prime}} ) \\{{{{A_{{X\; 1},k}(D)}{X_{1}(D)}} + {{A_{{X\; 2},k}(D)}{X_{2}(D)}} + \ldots + {{A_{{{Xy} - 1},k}(D)}{X_{y - 1}(D)}} + {{B_{k}(D)}{P(D)}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}( {k + 1} )} ) \\\begin{matrix}{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} \\{\mspace{79mu} \vdots}\end{matrix} & ( {{Equation}\mspace{14mu} 31\text{-}( {k + 1} )^{\prime}} ) \\{{{{A_{{X\; 1},{g - 1}}(D)}{X_{1}(D)}} + {{A_{{X\; 2},{g - 1}}(D)}{X_{2}(D)}} + \ldots + {{A_{{{Xy} - 1},{g - 1}}(D)}{X_{y - 1}(D)}} + {{B_{g - 1}(D)}{P(D)}}} = {0\mspace{14mu} ( {{g - 1} = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}g} ) \\{{{{B_{g - 1}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},{g - 1}}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {{g - 1} = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 31\text{-}g^{\prime}} )\end{matrix}$

In equations 31-1 to 31-g, equation 31-w and equation 31-w′ areequivalent, and therefore it is possible to replace equation 31-w belowwith equation 31-w′ (w=1, 2, g).

Then, as described above (with an LDPC-CC of good characteristics),information X₁, X₂, . . . , X_(y−1) at point in time j is represented asX_(1,j), X_(2,j), X_(y−1,j), parity P at point in time j is representedas P_(j), and u_(j)=(X_(i,j), X₂, X_(y−1,j), P_(j))^(T). At this time,information X_(1,j), X_(2,j), . . . , X_(y−1) and parity P_(j) at pointin time j: satisfy a parity check polynomial of equation 31-1 when “jmod g=0”; satisfy a parity check polynomial of equation 31-2 when “j modg=1”; satisfy a parity check polynomial of equation 31-3 when “j modg=2”; . . . ; satisfy a parity check polynomial of equation 31-(k+1)when “j mod g=k”; . . . ; and satisfy a parity check polynomial ofequation 31-g when “j mod g=g−1.” At this time, the relationshipsbetween parity check polynomials and a parity check matrix are the sameas above (i.e. as in an LDPC-CC of good characteristics).

Next, if w=z−1 is assumed in equations 30-w-0, 30-w-1, 30-w-2, . . . ,30-w-(g−2) and 30-w-(g−1), which represent g parity check polynomialsdescribed upon explaining equations 30-1 to 30-(q−1), representations ofg parity check polynomials can be provided as shown in equations 32-1 to32-g. Here, from the relationship of y<z≤q, representations of equations32-1 to 32-g can be provided.

$\begin{matrix}{\mspace{79mu} \lbrack 32\rbrack} & \; \\{{{{A_{{X\; 1},0}(D)}{X_{1}(D)}} + {{A_{{X\; 2},0}(D)}{X_{2}(D)}} + \ldots + {{A_{{{Xy} - 1},0}(D)}{X_{y - 1}(D)}} + \ldots + {{A_{{Xs},0}(D)}{X_{s}(D)}} + \ldots + {{A_{{{Xz} - 1},0}(D)}{X_{z - 1}(D)}} + {{B_{0}(D)}{P(D)}}} = {0\mspace{14mu} ( {0 = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}1} ) \\{{{{B_{0}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},0}(D)}{X_{r}(D)}}} + {\sum\limits_{s = y}^{z - 1}{{A_{{Xs},0}(D)}{X_{s}(D)}}}} = {0\mspace{14mu} ( {0 = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}1^{\prime}} ) \\{{{{A_{{X\; 1},1}(D)}{X_{1}(D)}} + {{A_{{X\; 2},1}(D)}{X_{2}(D)}} + \ldots + {{A_{{{Xy} - 1},1}(D)}{X_{y - 1}(D)}} + \ldots + {{A_{{Xs},1}(D)}{X_{s}(D)}} + \ldots + {{A_{{{Xz} - 1},1}(D)}{X_{z - 1}(D)}} + {{B_{1}(D)}{P(D)}}} = {0\mspace{14mu} ( {1 = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}2} ) \\{{{{B_{1}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},1}(D)}{X_{r}(D)}}} + {\sum\limits_{s = y}^{z - 1}{{A_{{Xs},1}(D)}{X_{s}(D)}}}} = {0\mspace{14mu} ( {1 = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}2^{\prime}} ) \\{\mspace{79mu} \vdots} & \; \\{{{{A_{{X\; 1},k}(D)}{X_{1}(D)}} + {{A_{{X\; 2},k}(D)}{X_{2}(D)}} + \ldots + {{A_{{{Xy} - 1},k}(D)}{X_{y - 1}(D)}} + \ldots + {{A_{{Xs},k}(D)}{X_{s}(D)}} + \ldots + {{A_{{{Xz} - 1},k}(D)}{X_{z - 1}(D)}} + {{B_{k}(D)}{P(D)}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}( {k + 1} )} ) \\{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}} + {\sum\limits_{s = y}^{z - 1}{{A_{{Xs},k}(D)}{X_{s}(D)}}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}( {k + 1} )^{\prime}} ) \\{\mspace{79mu} \vdots} & \; \\{{{{A_{{X\; 1},{g - 1}}(D)}{X_{1}(D)}} + {{A_{{X\; 2},{g - 1}}(D)}{X_{2}(D)}} + \ldots + {{A_{{{Xy} - 1},{g - 1}}(D)}{X_{y - 1}(D)}} + \ldots + {{A_{{Xs},{g - 1}}(D)}{X_{s}(D)}} + \ldots + {{A_{{{Xz} - 1},{g - 1}}(D)}{X_{z - 1}(D)}} + {{B_{g - 1}(D)}{P(D)}}} = {0\mspace{14mu} ( {{g - 1} = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}g} ) \\{{{{B_{g - 1}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},{g - 1}}(D)}{X_{r}(D)}}} + {\sum\limits_{s = y}^{z - 1}{{A_{{Xs},{g - 1}}(D)}{X_{s}(D)}}}} = {0\mspace{14mu} ( {{g - 1} = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 32\text{-}g^{\prime}} )\end{matrix}$

In equations 32-1 to 32-g, equation 32-w and equation 32-w′ areequivalent, and therefore it is possible to replace equation 32-w belowwith equation 32-w′ (w=1, 2, . . . , g).

Then, as described above (LDPC-CC of good characteristics), informationX₁, X₂, . . . , X_(y−1), . . . , X_(s), . . . , X_(z−1) at point in timej is represented as X_(1,j), X₂, . . . , X_(y−1,j), X_(s,j), . . . ,X_(z−1,j), parity P at point in time j is represented as P_(j), andu_(j)=(X_(1,j), X_(2,j), . . . , X_(y−1,j), . . . , X_(s,j), . . . ,X_(z−1,j), P_(j))^(T) (here, from the relationship of y<z≤q, s=y, y+1,y+2, y+3, . . . z−3, z−2, z−1). At this time, information X_(1,j),X_(2,j), . . . , X_(y−1,j), . . . , X_(s,j), . . . X_(z−1,j) and parityP_(j) at point in time j: satisfy a parity check polynomial of equation32-1 when “j mod g=0”; satisfy a parity check polynomial of equation32-2 when “j mod g=1”; satisfy a parity check polynomial of equation32-3 when “j mod g=2”; . . . , satisfy a parity check polynomial ofequation 32-(k+1) when “j mod g=k”; . . . ; and satisfy a parity checkpolynomial of equation 32-g when “j mod g=g−1.” At this time, therelationships between parity check polynomials and a parity check matrixare the same as above (i.e. as in an LDPC-CC of good characteristics).

In a case where the above relationships hold true, if the followingconditions hold true for an LDPC-CC of a time varying period of g in acoding rate of (y−1)/y and for an LDPC-CC of a time varying period of gin a coding rate of (z−1)/z, it is possible to share circuits between anencoder for an LDPC-CC of a time varying period of g in a coding rate of(y−1)/y and an encoder for an LDPC-CC of a time varying period of g in acoding rate of (z−1)/z, and it is possible to share circuits between adecoder for an LDPC-CC of a time varying period of g in a coding rate of(y−1)/y and a decoder for an LDPC-CC of a time varying period of g in acoding rate of (z−1)/z. The conditions are as follows.

First, the following relationships hold true between equation 31-1 andequation 32-1:

A _(X1,0)(D) of equation 31-1 and A _(X1,0)(D) of equation 32-1 areequal;

A _(Xf,0)(D) of equation 31-1 and A _(Xf,0)(D) of equation 32-1 areequal;

A _(Xy−1,0)(D) of equation 31-1 and A _(Xy−1,0)(D) of equation 32-1 areequal. That is, the above relationships hold true for f=1,2,3, . . .,y−1.

Also, the following relationship holds true for parity:

B ₀(D) of equation 31-1 and B ₀(D) of equation 32-1 are equal.

Similarly, the following relationships hold true between equation 31-2and equation 32-2:

A _(X1,1)(D) of equation 31-2 and A _(X1,1)(D) of equation 32-2 areequal;

A _(Xf,1)(D) of equation 31-2 and A _(Xf,1)(D) of equation 32-2 areequal;

A _(Xy−1,1)(D) of equation 31-2 and A _(Xy−1,1)(D) of equation 32-2 areequal. That is, the above relationships hold true for f=1,2,3, . . .,y−1.

Also, the following relationship holds true for parity:

B ₁(D) of equation 31-2 and B ₁(D) of equation 32-2 are equal, and soon.

Similarly, the following relationships hold true between equation 31-hand equation 32-h:

A _(X1,h−1)(D) of equation 31-h and A _(X1,h−1)(D) of equation 32-h areequal;

A _(Xf,h−1)(D) of equation 31-h and A _(Xf,h−1)(D) of equation 32-h areequal;

A _(Xy−i,h−i)(D) of equation 31-h and A _(Xy−i,h−i)(D) of equation 32-hare equal.

That is, the above relationships hold true for f=1, 2, 3, . . . , y−1.

Also, the following relationship holds true for parity:

B _(h−1)(D) of equation 31-h and B _(h−1)(D) of equation 32-h are equal,and so on.

Similarly, the following relationships hold true between equation 31-gand equation 32-g:

A _(X1,g−1)(D) of equation 31-g and A _(X1,g−1)(D) of equation 32-g areequal;

A _(Xf,g−1)(D) of equation 31-g and A _(Xf,g−1)(D) of equation 32-g areequal;

A _(Xy−1,g−1)(D) of equation 31-g and A _(Xy−1,g−1)(D) of equation 32-gare equal.

That is, the above relationships hold true for f=1, 2, 3, . . . , y−1.

Also, the following relationship holds true for parity:

B _(g−1)(D) of equation 31-g and B _(g−1)(D) of equation 32-g are equal(therefore, h=1,2,3, . . . ,g−2,g−1,g).

In a case where the above relationships hold true, it is possible toshare circuits between an encoder for an LDPC-CC of a time varyingperiod of g in a coding rate of (y−1)/y and an encoder for an LDPC-CC ofa time varying period of g in a coding rate of (z−1)/z, and it ispossible to share circuits between a decoder for an LDPC-CC of a timevarying period of g in a coding rate of (y−1)/y and a decoder for anLDPC-CC of a time varying period of g in a coding rate of (z−1)/z. Here,the method of sharing encoder circuits and the method of sharing decodercircuits will be explained in detail in the following (configurations ofan encoder and decoder).

Examples of LDPC-CC parity check polynomials will be shown in table 5,where the time varying period is 3 and the coding rate is 1/2, 2/3, 3/4or 5/6. Here, the form of parity check polynomials is the same as in theform of table 3. By this means, if the transmitting apparatus and thereceiving apparatus support coding rates of 1/2, 2/3, 3/4 and 5/6 (or ifthe transmitting apparatus and the receiving apparatus support two ormore of the four coding rates), it is possible to reduce thecomputational complexity (circuit scale) (this is because it is possibleto share encoder circuits and decoder circuits even in the case ofdistributed codes, and therefore reduce the circuit scale), and providedata of high received quality in the receiving apparatus.

TABLE 5 Code Parity check polynomial LDPC- Check polynomial #1: (D³⁷³ +D⁵⁶ + 1)X₁(D) + (D⁴⁰⁶ + D²¹⁸ + 1)P(D) = 0 CC of a Check polynomial #2:(D⁴⁵⁷ + D¹⁹⁷ + 1)X₁(D) + (D⁴⁹¹ + D²² + 1)P(D) = 0 time Check polynomial#3: (D⁴⁸⁵ + D⁷⁰ + 1)X₁(D) + (D²³⁶ + D¹⁸¹ + 1)P(D) = 0 varying period of3 and a coding rate of 1/2 LDPC- Check polynomial #1: CC of a (D³⁷³ +D⁵⁶ + 1)X₁(D) + (D⁸⁶ + D⁴ + 1)X₂(D) + (D⁴⁰⁶ + D²¹⁸ + 1)P(D) = 0 timeCheck polynomial #2: varying (D⁴⁵⁷ + D¹⁹⁷ + 1)X₁(D) + (D³⁶⁸ + D²⁹⁵ +1)X₂(D) + (D⁴⁹¹ + D²² + 1)P(D) = 0 period Check polynomial #3: of 3 and(D⁴⁸⁵ + D⁷⁰ + 1)X₁(D) + (D⁴⁷⁵ + D³⁹⁸ + 1)X₂(D) + (D²³⁶ + D¹⁸¹ + 1)P(D) =0 a coding rate of 2/3 LDPC- Check polynomial #1: (D³⁷³ + D⁵⁶ +1)X₁(D) + (D⁸⁶ + D⁴ + 1)X₂(D) + CC of a (D³⁸⁸ + D¹³⁴ + 1)X₃(D) + (D⁴⁰⁶ +D²¹⁸ + 1)P(D) = 0 time Check polynomial #2: (D⁴⁵⁷ + D¹⁹⁷ + 1)X₁(D) +(D³⁶⁸ + D²⁹⁵ + 1)X₂(D) + varying (D¹⁵⁵ + D¹³⁶ + 1)X₃(D) + (D⁴⁹¹ + D²² +1)P(D) = 0 period Check polynomial #3: (D⁴⁸⁵ + D⁷⁰ + 1)X₁(D) + (D⁴⁷⁵ +D³⁹⁸ + 1)X₂(D) + of 3 and (D⁴⁹³ + D⁷⁷ + 1)X₃(D) + (D²³⁶ + D¹⁸¹ + 1)P(D)= 0 a coding rate of 3/4 LDPC- Check polynomial #1: CC of a (D³⁷³ +D⁵⁶ + 1)X₁(D) + (D⁸⁶ + D⁴ + 1)X₂(D) + (D³⁸⁸ + D¹³⁴ + 1)X₃(D) + time(D²⁵⁰ + D¹⁹⁷ + 1)X₄(D) + (D²⁹⁵ + D¹¹³ + 1)X₅(D) + (D⁴⁰⁶ + D²¹⁸ + 1)P(D)= 0 varying Check polynomial #2: period (D⁴⁵⁷ + D¹⁹⁷D + 1)X₁(D) +(D³⁶⁸ + D²⁹⁵ + 1)X₂(D) + (D¹⁵⁵ + D¹³⁶ + 1)X₃(D) + of 3 and (D²²⁰ +D¹⁴⁶ + 1)X₄(D) + (D³¹¹ + D¹¹⁵ + 1)X₅(D) + (D⁴⁹¹ + D²² + 1)P(D) = 0 aCheck polynomial #3: coding (D⁴⁸⁵ + D⁷⁰ + 1)X₁(D) + (D⁴⁷⁵ + D³⁹⁸ +1)X₂(D) + (D⁴⁹³ + D⁷⁷ + 1)X₃(D) + rate of (D⁴⁹⁰ + D²³⁹ + 1)X₄(D) +(D³⁹⁴ + D²⁷⁸ + 1)X₅(D) + (D²³⁶+ D¹⁸¹ + 1)P(D) = 0 5/6

A case will be explained where LDPC-CCs of a time varying period of 3 intable 5 satisfy the above conditions. For example, consider an LDPC-CCof a time varying period of 3 in a coding rate of 1/2 in table 5 and anLDPC-CC of a time varying period of 3 in a coding rate of 2/3 in table5. That is, y=2 holds true in equations 31-1 to 31-g, and z=3 holds truein equations 32-1 to 32-g.

Then, seen from an LDPC-CC of a time varying period of 3 in a codingrate of 1/2 in table 5, A_(X1,0)(D) of equation 31-1 representsD³⁷³+D⁵⁶+1, and, seen from an LDPC-CC of a time varying period of 3 in acoding rate of 2/3 in table 5, A_(X1,0)(D) of equation 32-1 representsD³⁷³+D⁵⁶+1, so that A_(X1,0)(D) of equation 31-1 and A_(X1,0)(D) ofequation 32-1 are equal.

Also, seen from an LDPC-CC of a time varying period of 3 in a codingrate of 1/2 in table 5, B₀(D) of equation 31-1 represents D⁴⁰⁶+D²¹⁸+1,and, seen from an LDPC-CC of a time varying period of 3 in a coding rateof 2/3 in table 5, B₀(D) of equation 32-1 represents D⁴⁰⁶⁺¹)²¹⁸+1, sothat B₀(D) of equation 31-1 and B₀(D) of equation 32-1 are equal.

Similarly, seen from an LDPC-CC of a time varying period of 3 in acoding rate of 1/2 in table 5, A_(X1,1)(D) of equation 31-2 representsD⁴⁵⁷+D¹⁹⁷+1, and, seen from an LDPC-CC of a time varying period of 3 ina coding rate of 2/3 in table 5, A_(X1,1)(D) of equation 32-2 representsD⁴⁵⁷+D¹⁹⁷+1, so that A_(X1,1)(D) of equation 31-2 and A_(X1,1)(D) ofequation 32-2 are equal.

Also, seen from an LDPC-CC of a time varying period of 3 in a codingrate of 1/2 in table 5, B_(i)(D) of equation 31-2 represents D⁴⁹¹+D²²+1,and, seen from an LDPC-CC of a time varying period of 3 in a coding rateof 2/3 in table 5, B_(i)(D) of equation 32-2 represents D⁴⁹¹+D²²+1, sothat B_(i)(D) of equation 31-2 and B_(i)(D) of equation 32-2 are equal.

Similarly, seen from an LDPC-CC of a time varying period of 3 in acoding rate of 1/2 in table 5, A_(X1,2)(D) of equation 31-3 representsD⁴⁸⁵+D⁷⁰+1, and, seen from an LDPC-CC of a time varying period of 3 in acoding rate of 2/3 in table 5, A_(X1,2)(D) of equation 32-3 representsD⁴⁸⁵+D⁷⁰+1, so that A_(X1,2)(D) of equation 31-3 and A_(X1,2)(D) ofequation 32-3 are equal.

Also, seen from an LDPC-CC of a time varying period of 3 in a codingrate of 1/2 in table 5, B₂(D) of equation 31-3 represents D²³⁶+D¹⁸¹+1,and, seen from an LDPC-CC of a time varying period of 3 in a coding rateof 2/3 in table 5, B₂(D) of equation 32-3 represents D236+D¹⁸¹+1, sothat B₂(D) of equation 31-3 and B₂(D) of equation 32-3 are equal.

In view of the above, it is confirmed that an LDPC-CC of a time varyingperiod of 3 in a coding rate of 1/2 in table 5 and an LDPC-CC of a timevarying period of 3 in a coding rate of 2/3 in table 5 satisfy the aboveconditions.

Similarly as above, if LDPC-CCs of a time varying period of 3 in twodifferent coding rates are selected from LDPC-CCs of a time varyingperiod of 3 in coding rates of 1/2, 2/3, 3/4 and 5/6 in table 5, andwhether or not the above conditions are satisfied is examined, it isconfirmed that the above conditions are satisfied in any selectedpatterns.

Also, an LDPC-CC is a kind of a convolutional code, and thereforerequires, for example, termination or tail-biting to secure belief indecoding of information bits. Here, a case will be considered where themethod of making the state of data (information) X zero (hereinafter“information-zero-termination”) is implemented.

FIG. 10 shows the method of information-zero-termination. As shown inFIG. 10, the information bit (final transmission bit) that is finallytransmitted in a transmission information sequence is Xn(110). With thisfinal information bit Xn(110), if the transmitting apparatus transmitsdata only until parity bits generated in an encoder and then thereceiving apparatus performs decoding, the received quality ofinformation degrades significantly. To solve this problem, informationbits subsequent to final information bit Xn(110) (hereinafter “virtualinformation bits”) are presumed as “0” and encoded to generate paritybits 130.

In this case, the receiving apparatus knows that virtual informationbits 120 are “0,” so that the transmitting apparatus does not transmitvirtual information bits 120, but transmits only parity bits 130generated by virtual information bits 120 (these parity bits representredundant bits that need to be transmitted, and therefore are called“redundant bits”). Then, a new problem arises that, in order to enableboth improvement of efficiency of data transmission and maintenance ofreceived quality of data, it is necessary to secure the received qualityof data and decrease the number of parity bits 130 generated by virtualinformation bits 120 as much as possible.

At this time, it is confirmed by simulation that, in order to secure thereceived quality of data and decrease the number of parity bitsgenerated by virtual information bits, terms related to parity of aparity check polynomial play an important role.

As an example, a case will be explained using an LDPC-CC for which thetime varying period is m (where m is an integer equal to or greater than2) and the coding rate is 1/2. When the time varying period is m, mnecessary parity check polynomials are represented by the followingequation.

[33]

A _(X1,i)(D)X ₁(D)+B _(i)(D)P(D)=0  (Equation 33)

where i=0, 1, . . . , m−1. Also, assume that all of the orders of D inA_(X1,1)(D) are integers equal to or greater than 0 (e.g. as shown inA_(X1,1)(D)=D¹⁵+D³+D⁰, the orders of D are 15, 3 and 0, all of which areintegers equal to or greater than 0), and all of the orders of D inB₁(D) are also integers equal to or greater than 0 (e.g. as shown inB_(i)(D)=D¹⁸+D⁴±D⁰, the orders of D are 18, 4 and 0, all of which areintegers equal to or greater than 0).

Here, at time j, the parity check polynomial of the following equationholds true.

[34]

A _(X1,k)(D)X ₁(D)+B _(k)(D)P(D)=0(k=j mod m)  (Equation 34)

Then, in X₁(D), assume that: the highest order of D in A_(X1,1)(D) is α₁(e.g. when A_(X1,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α₁); the highest orderof D in A_(X1,2)(D) is α₂; . . . ; the highest order of D in A_(X1,1)(D)is α_(i); . . . ; and the highest order of D in A_(X1,m−1)(D) isα_(m−1). Then, the highest value in α_(i) (where i=0, 1, 2, . . . , m−1)is α.

On the other hand, in P(D), assume that: the highest order of D in B₁(D)is β₁; the highest order of D in B₂(D) is β₂; . . . ; the highest orderof D in B_(i)(D) is β_(i); . . . ; and the highest order of D inB_(m−1)(D) is β_(m−1). Then, the highest value in β_(i) (where i=0, 1,2, . . . , m−1) is β.

Then, in order to secure the received quality of data and decrease thenumber of parity bits generated by virtual information bits as much aspossible, it is preferable to set β equal to or below half of α.

Although a case has been described where the coding rate is 1/2, thesame applies to other cases where the coding rate is above 1/2. At thistime, especially when the coding rate is equal to or greater than 4/5,there is a trend to require a significant number of redundant bits tosatisfy conditions for securing the received quality of data anddecreasing the number of parity bits generated by virtual informationbits as much as possible. Consequently, the conditions described aboveplay an important role to secure the received quality of data anddecrease the number of parity bits generated by virtual information bitsas much as possible.

As an example, a case will be explained using an LDPC-CC for which thetime varying period is m (where m is an integer equal to or greater than2) and the coding rate is 4/5. When the time varying period is m, mnecessary parity check polynomials are represented by the followingequation.

[35]

A _(X1,i)(D)X ₁(D)+A _(X2,i)(D)X ₂(D)+A _(X3,i)(D)X ₃(D)+A _(X4,i)(D)X₄(D)+B _(i)(D)+B _(i)(D)P(D)=0  (Equation 35)

where i=0, 1, . . . , m−1. Also, assume that all of the orders of D inA_(X1,i)(D) are integers equal to or greater than 0 (e.g. as shown inA_(X1,1)(D)=D¹⁵+D³+D⁰, the orders of D are 15, 3 and 0, all of which areintegers equal to or greater than 0). Similarly, assume that: all of theorders of D in A_(X2,i)(D) are integers equal to or greater than 0; allof the orders of D in A_(X3,i)(D) are integers equal to or greater than0; all of the orders of D in A_(X4,i)(D) are integers equal to orgreater than 0; and all of the orders of D in B₁(D) are integers equalto or greater than 0 (e.g. as shown in B_(i)(D)=D¹⁸+D⁴+D⁰, the orders ofD are 18, 4 and 0, all of which are integers equal to or greater than0).

Here, at time j, the parity check polynomial of the following equationholds true.

[36]

A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+A _(X3,k)(D)X ₃(D)+A _(X4,k)(D)X₄₁(D)+B _(k)(D)P(D)=0(k=j mod m)  (Equation 35)

Then, in X₁(D), assume that: the highest order of D in A_(X1,1)(D) isα_(1,1) (e.g. when A_(X1,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and0, and therefore provides 15 as the highest order of D, α_(1,1)); thehighest order of D in A_(X1,2)(D) is α_(1,2); . . . ; the highest orderof D in A_(X1,i)(D) is α_(1,i); . . . ; and the highest order of D inA_(X1,m−1)(D) is α_(1,m−1). Then, the highest value in α_(1,i) (wherei=0, 1, 2, . . . , m−1) is α₁.

In X₂(D), assume that: the highest order of D in A_(X2,1)(D) is α_(2,1)(e.g. when A_(X2,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α_(2,1)); the highestorder of D in A_(X2,2)(D) is α_(2,2); . . . ; the highest order of D inA_(X2,i)(D) is α_(2,i); . . . ; and the highest order of D inA_(X2,m−1)(D) is α_(2,m−1). Then, the highest value in α_(2,i) (wherei=0, 1, 2, . . . , m−1) is α₂.

In X₃(D), assume that: the highest order of D in A_(X3,1)(D) is α_(3,1)(e.g. when A_(X3,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α_(3,1)); the highestorder of D in A_(X3,2)(D) is α_(3,2); . . . ; the highest order of D inA_(X3,i)(D) is α_(3,i); . . . ; and the highest order of D inA_(X3,m−1)(D) is α_(3,m−1). Then, the highest value in α_(3,i) (wherei=0, 1, 2, . . . , m−1) is α₃.

In X₄(D), assume that: the highest order of D in A_(X4,1)(D) is α_(4,1)(e.g. when A_(X4,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α_(4,i)); the highestorder of D in A_(X4,2)(D) is α_(4,2), . . . ; the highest order of D inA_(X4,i)(D) is α_(4,i); . . . ; and the highest order of D inA_(X4,m−1)(D) is α_(4,m−1). Then, the highest value in α_(4,i) (wherei=0, 1, 2, . . . , m−1) is α₄.

In P(D), assume that: the highest order of D in B₁(D) is β₁; the highestorder of D in B₂(D) is β₂; . . . ; the highest order of D in B_(i)(D) isρ_(i); . . . ; and the highest order of D in B_(m−1)(D) is β_(m−1).Then, the highest value in β_(i) (where i=0, 1, 2, . . . , m−1) is β.

Then, in order to secure the received quality of data and decrease thenumber of parity bits generated by virtual information bits as much aspossible, it is necessary to satisfy conditions that: β is equal to orbelow half of α₁; β is equal to or below half of α₂; β is equal to orbelow half of α₃; and β is equal to or below half of α₄, so that,especially, there is a high possibility to secure the received qualityof data.

Also, even in a case where: β is equal to or below half of α₁; β isequal to or below half of α₂; β is equal to or below half of α₃; or β isequal to or below half of α₄, although it is possible to secure thereceived quality of data and decrease the number of parity bitsgenerated by virtual information bits as much as possible, there is alittle possibility to cause degradation in the received quality of data(here, degradation in the received quality of data is not necessarilycaused).

Therefore, in the case of an LDPC-CC for which the time varying periodis m (where m is an integer equal to or greater than 2) and the codingrate is (n−1)/n, the following is possible.

When the time varying period is m, m necessary parity check polynomialsare represented by the following equation.

[37]

A _(X1,i)(D)X ₁(D)+A _(X2,i)(D)X ₂(D)+ . . . +A _(Xn−1,i)(D)X_(n−1)(D)+B _(i)(D)P(D)=0  (Equation 37)

where i=0, 1, . . . , m−1. Also, assume that all of the orders of D inA_(X1,i)(D) are integers equal to or greater than 0 (e.g. as shown inA_(X1,1)(D)=D¹⁵+D³+D⁰, the orders of D are 15, 3 and 0, all of which areintegers equal to or greater than 0). Similarly, assume that: all of theorders of D in A_(X2,i)(D) are integers equal to or greater than 0; allof the orders of D in A_(X3,i)(D) are integers equal to or greater than0; all of the orders of D in A_(X4,i)(D) are integers equal to orgreater than 0; . . . ; all of the orders of D in A_(Xu,i)(D) areintegers equal to or greater than 0; . . . ; all of the orders of D inA_(Xn−1,i)(D) are integers equal to or greater than 0; and all of theorders of D in B₁(D) are integers equal to or greater than 0 (e.g. asshown in B_(i)(D)=D¹⁸+D⁴+D⁰, the orders of D are 18, 4 and 0, all ofwhich are integers equal to or greater than 0).

Here, at time j, the parity check polynomial of the following equationholds true.

[38]

A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+ . . . +A _(Xn−1,k)(D)X_(n−1)(D)+B _(k)(D)P(D)=0(k=j mod m)  (Equation 38)

Then, in X₁(D), assume that: the highest order of D in A_(X1,1)(D) isα_(1,1) (e.g. when A_(X1,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and0, and therefore provides 15 as the highest order of D, α_(1,i)); thehighest order of D in A_(X1,2)(D) is α_(1,2); . . . ; the highest orderof D in A_(X1,i)(D) is α_(1,i); . . . ; and the highest order of D inA_(X1,m−1)(D) is α_(1, m−1). Then, the highest value in α_(1,i) (wherei=0, 1, 2, . . . , m−1) is α₁.

In X₂(D), assume that: the highest order of D in A_(X2,1)(D) is α_(2,1)(e.g. when A_(X2,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and 0, andtherefore provides 15 as the highest order of D, α_(2,1)); the highestorder of D in A_(X2,2)(D) is α_(2,2); . . . ; the highest order of D inA_(X2,i)(D) is a_(2,i); . . . ; and the highest order of D inA_(X2,m−1)(D) is α_(2,m−1). Then, the highest value in α_(2,1) (wherei=0, 1, 2, . . . , m−1) is α₂.

In X_(u)(D), assume that: the highest order of D in A_(Xu,1)(D) isα_(u,1) (e.g. when A_(Xu,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3 and0, and therefore provides 15 as the highest order of D, α_(u,1)); thehighest order of D in A_(Xu,m−1)(D) is α_(u,2); . . . ; the highestorder of D in A_(Xu,i)(D) is α_(u,i); . . . ; and the highest order of Din A_(Xu,m−1)(D) is α_(u,m−1). Then, the highest value in α_(u,i) (wherei=0, 1, 2, . . . , m−1, u=1, 2, 3, . . . , n−2, n−1) is α_(u).

In X_(n−1)(D), assume that: the highest order of D in A_(Xn−1,1)(D) isα_(n−1,1) (e.g. when A_(Xn−1,1)(D)=D¹⁵+D³+D⁰, D has the orders of 15, 3and 0, and therefore provides 15 as the highest order of D, α_(n−1,1));the highest order of D in A_(Xn−1,2)(D) is α_(n−1,2); . . . ; thehighest order of D in A_(Xn−1,i)(D) is α_(n−1,i); . . . ; and thehighest order of D in A_(Xn−1,m−1)(D) is α_(n−1,m−1). Then, the highestvalue in α_(n−1,i) (where i=0, 1, 2, . . . , m−1) is α_(n−1).

In P(D), assume that: the highest order of D in B₁(D) is β₁; the highestorder of D in B₂(D) is β₂; . . . ; the highest order of D in B_(i)(D) isβ_(i); . . . ; and the highest order of D in B_(m−1)(D) is β_(m−1).Then, the highest value in β_(i) (where i=0, 1, 2, . . . , m−1) is β.

Then, in order to secure the received quality of data and decrease thenumber of parity bits generated by virtual information bits as much aspossible, it is necessary to satisfy conditions that: β is equal to orbelow half of α₁; β is equal to or below half of α₂; . . . ; β is equalto or below half of α_(u); . . . ; and β is equal to or below half ofα_(n−1) (where u=1, 2, 3, . . . , n−2, n−1), so that, especially, thereis a high possibility to secure the received quality of data.

Also, even in a case where: β is equal to or below half of α₁; β isequal to or below half of α₂; . . . ; β is equal to or below half ofα_(u); . . . ; or β is equal to or below half of α_(n−1) (where u=1, 2,3, . . . , n−2, n−1), although it is possible to secure the receivedquality of data and decrease the number of parity bits generated byvirtual information bits as much as possible, there is a littlepossibility to cause degradation in the received quality of data (here,degradation in the received quality of data is not necessarily caused).

Table 6 shows an example of LDCPC-CC parity check polynomials that cansecure the received quality of data and reduce redundant bits, where thetime varying period is 3 and the coding rate is 1/2, 2/3, 3/4 or 4/5. IfLDPC-CCs of a time varying period of 3 in two different coding rates areselected from LDPC-CCs of a time varying period of 3 in coding rates of1/2, 2/3, 3/4 and 4/5 in table 6, and whether or not the above-describedconditions for sharing encoders and decoders are satisfied is examined,similar to LDPC-CCs of a time varying period of 3 in table 5, it isconfirmed that the above conditions for enabling sharing process inencoders and decoders are satisfied in any selected patterns.

Also, although 1000 redundant bits are required in a coding rate of 5/6in table 5, it is confirmed that the number of redundant bits is equalto or below 500 bits in a coding rate of 4/5 in table 6.

Also, in the codes of table 6, the number of redundant bits (which areadded for information-zero-termination) varies between coding rates. Atthis time, the number of redundant bits tends to increase when thecoding rate increases. That is, in a case where codes are created asshown in tables 5 and 6, if there are a code of a coding rate of (n−1)/nand a code of a coding rate of (m−1)/m (n>m), the number of redundantbits required for the code of a coding rate of (n−1)/n (i.e. the numberof redundant bits added for information-zero-termination) is more thanthe number of redundant bits required for the code of a coding rate of(m−1)/m (i.e. the number of redundant bits added forinformation-zero-termination).

TABLE 6 Code Parity check polynomial LDPC-CC of Check polynomial #1: atime (D²⁶⁸ + D¹⁶⁴ + 1)X₁(D) + (D⁹² + D⁷ + 1)P(D) = 0 varying Checkpolynomial #2: period of 3 (D³⁷⁰ + D³¹⁷ + 1)X₁(D) + (D⁹⁵ + D²² + 1)P(D)= 0 and a coding Check polynomial #3: rate of 1/2 (D³⁴⁶ + D⁸⁶ +1)X₁(D) + (D⁸⁸ + D²⁶ + 1)P(D) = 0 LDPC-CC of Check polynomial #1: a time(D²⁶⁸ + D¹⁶⁴ + 1)X₁(D) + (D³⁸⁵ + D²⁴² + 1) varying X₂(D) + (D⁹² + D⁷ +1)P(D) = 0 period of 3 Check polynomial #2: and a coding (D³⁷⁰ + D³¹⁷ +1)X₁(D) + (D¹²⁵ + D¹⁰³ + 1) rate of 2/3 X₂(D) + (D⁹⁵ + D²² + 1)P(D) = 0Check polynomial #3: (D³⁴⁶ + D⁸⁶ + 1)X₁(D) + (D³¹⁹ + D²⁹⁰ + 1) X₂(D) +(D⁸⁸ + D²⁶ + 1)P(D) = 0 LDPC-CC of Check polynomial #1: (D²⁶⁸ + D¹⁶⁴ +1)X₁(D) + a time (D³⁸⁵ + D²⁴² + 1)X₂(D) + varying (D³⁴³ + D²⁸⁴ +1)X₃(D) + (D⁹² + D⁷ + 1)P(D) = 0 period of 3 Check polynomial #2:(D³⁷⁰ + D³¹⁷ + 1)X₁(D) + and a coding (D¹²⁵ + D¹⁰³ + 1)X₂(D) + rate of3/4 (D²⁵⁹ + D¹⁴ + 1)X₃(D) + (D⁹⁵ + D²² + 1)P(D) = 0 Check polynomial #3:(D³⁴⁶ + D⁸⁶ + 1)X₁(D) + (D³¹⁹ + D²⁹⁰ + 1)X₂(D) + (D¹⁴⁵ + D¹¹ + 1)X₃(D) +(D⁸⁸ + D²⁶ + 1)P(D) = 0 LDPC-CC of Check polynomial #1: a time (D²⁶⁸ +D¹⁶⁴ + 1)X₁(D) + (D³⁸⁵ + D²⁴² + 1)X₂(D) + varying (D³⁴³ + D²⁸⁴ +1)X₃(D) + (D³¹⁰ + D¹¹³ + 1) period of 3 X₄(D) + (D⁹² + D⁷ + 1)P(D) = 0and a coding Check polynomial #2: rate of 5/6 (D³⁷⁰ + D³¹⁷ + 1)X₁(D) +(D¹²⁵ + D¹⁰³ + 1)X₂(D) + (D²⁵⁹ + D¹⁴ + 1)X₃(D) + (D³⁹⁴ + D¹⁸⁸ + 1)X₄(D) + (D⁹⁵ + D²² + 1)P(D) = 0 Check polynomial #3: (D³⁴⁶ + D⁸⁶ +1)X₁(D) + (D³¹⁹ + D²⁹⁰ + 1)X₂(D) + (D¹⁴⁵ + D¹¹ + 1)X₃(D) + (D²³⁹ +D⁶⁷ + 1) X₄(D) + (D⁸⁸ + D²⁶ + 1)P(D) = 0

A case has been described above where the maximum coding rate is (q−1)/qamong coding rates of enabling encoder circuits to be shared andenabling decoder circuits to be shared, and where an LDPC-CC paritycheck polynomial of a coding rate of (r−1)/r (r=2, 3, . . . , q (q is anatural number equal to or greater than 3)) and a time varying period ofg is provided.

Here, the method of generating an LDPC-CC parity check polynomial of atime varying period of g for reducing the computational complexity (i.e.circuit scale) in a transmitting apparatus and receiving apparatus, andfeatures of parity check polynomials have been described, where thetransmitting apparatus provides at least an LDPC-CC encoder of a codingrate of (y−1)/y and a time varying period of g and an LDPC-CC encoder ofa coding rate of (z−1)/z (y≠z) and a time varying period of g, and wherethe receiving apparatus provides at least an LDPC-CC decoder of a codingrate of (y−1)/y and a time varying period of g and an LDPC-CC decoder ofa coding rate of (z−1)/z (y≠z) and a time varying period of g.

Here, the transmitting apparatus refers to a transmitting apparatus thatcan generate at least one of a modulation signal for transmitting anLDPC-CC coding sequence of a coding rate of (y−1)/y and a time varyingperiod of g and an LDPC-CC coding sequence of a coding rate of (z−1)/zand a time varying period of g.

Also, the receiving apparatus refers to a receiving apparatus thatdemodulates and decodes at least one of a received signal including anLDPC-CC coding sequence of a coding rate of (y−1)/y and a time varyingperiod of g and a received signal including an LDPC-CC coding sequenceof a coding rate of (z−1)/z and a time varying period of g.

By using an LDPC-CC of a time varying period of g proposed by thepresent invention, it is possible to provide an advantage of reducingthe computational complexity (i.e. circuit scale) in a transmittingapparatus including encoders and in a receiving apparatus includingdecoders (i.e. it is possible to share circuits).

Further, by using an LDPC-CC of a time varying period of g proposed bythe present invention, it is possible to provide an advantage ofacquiring data of high received quality in the receiving apparatus inany coding rates. Also, the configurations of encoders and decoders, andtheir operations will be described later in detail.

Also, although a case has been described above where LDPC-CCs of a timevarying period of g in coding rates of 1/2, 2/3, 3/4, . . . , and(q−1)/q are provided in equations 30-1 to 30-(q−1), a transmittingapparatus including encoders and a receiving apparatus includingdecoders need not support all of the coding rates of 1/2, 2/3, 3/4, . .. , and (q−1)/q. That is, as long as these apparatuses support at leasttwo or more different coding rates, it is possible to provide anadvantage of reducing the computational complexity (or circuit scale) inthe transmitting apparatus and the receiving apparatus (i.e. sharingencoder circuits and decoder circuits), and acquiring data of highreceived quality in the receiving apparatus.

Also, if all of coding rates supported by the transmitting and receivingapparatuses (encoders/decoders) are associated with codes based on themethods described with the present embodiment, by providingencoders/decoders of the highest coding rate among the supported codingrates, it is possible to easily support coding and decoding in allcoding rates and, at this time, provide an advantage of reducing thecomputational complexity significantly.

Also, although a case has been described above based on codes (i.e.LDPC-CCs of good characteristics), it is not necessary to satisfy theconditions described with the LDPC-CCs of good characteristics. That is,as long as LDPC-CCs of a time varying period of g are provided based onparity check polynomials in the form described with the LDPC-CCs of goodcharacteristics, it is equally possible to implement the presentembodiment (where g is an integer equal to or greater than 2). This isobvious from the relationships between equations 31-1 to 31-g andequations 32-1 to 32-g.

Naturally, for example, in a case where: the transmitting and receivingapparatuses (encoders/decoders) support coding rates of 1/2, 2/3, 3/4and 5/6; LDPC-CCs based on the above conditions are used in coding ratesof 1/2, 2/3 and 3/4; and codes not based on the above conditions areused in a coding rate of 5/6, it is possible to share circuits in theencoders and decoders in coding rates of 1/2, 2/3 and 3/4, and it isdifficult to share circuits in these encoders and decoders to sharecircuits in a coding rate of 5/6.

Embodiment 2

Embodiment 2 will explain in detail the method of sharing encodercircuits of an LDPC-CC formed by the search method explained inEmbodiment 1 and the method of sharing decoder circuits of that LDPC-CC.

First, in a case where the highest coding rate is (q−1)/q among codingrates for sharing encoder circuits and decoder circuits, an LDPC-CCencoder of a time varying rate of g (where g is a natural number)supporting a plurality of coding rates, (r−1)/r, will be explained(e.g., when the coding rates supported by a transmitting and receivingapparatus are 1/2, 2/3, 3/4 and 5/6, coding rates of 1/2, 2/3 and 3/4allow the circuits of encoders/decoders to be shared, while a codingrate of 5/6 does not allow the circuits of encoders/decoders to beshared, where the above highest coding rate, (q−1)/q, is 3/4).

FIG. 11 is a block diagram showing an example of the main components ofan encoder according to the present embodiment. Also, encoder 200 shownin FIG. 11 refers to an encoder that can support coding rates of 1/2,2/3 and 3/4. Encoder 200 shown in FIG. 11 is mainly provided withinformation generating section 210, first information computing section220-1, second information computing section 220-2, third informationcomputing section 220-3, parity computing section 230, adding section240, coding rate setting section 250 and weight control section 260.

Information generating section 210 sets information X_(1,i), informationX_(2,i) and information X_(3,i) at point in time i, according to acoding rate designated from coding rate setting section 250. Forexample, if coding rate setting section 250 sets the coding rate to 1/2,information generating section 210 sets information X_(1,i) at point intime i to input information data S_(j), and sets information X_(2,i) andinformation X_(3,i) at point in time i to “0.”

Also, in the case of a coding rate of 2/3, information generatingsection 210 sets information X_(1,i) at point in time i to inputinformation data S_(j), sets information X_(2,i) at point in time i toinput information data S_(j+1) and sets information X_(3,i) at point intime i to “0.”

Also, in the case of a coding rate of 3/4, information generatingsection 210 sets information X_(1,i) at point in time i to inputinformation data S_(j), sets information X_(2,i) at point in time i toinput information data S_(j+1) and sets information X_(3,i) at point intime i to input information data S_(j+2).

In this way, using input information data, information generatingsection 210 sets information X_(1,i), information X_(2,i) andinformation X_(3,i) at point in time i according to a coding rate set incoding rate setting section 250, outputs set information X_(1,i) tofirst information computing section 220-1, outputs set informationX_(2,i) to second information computing section 220-2 and outputs setinformation X_(3,i) to third information computing section 220-3.

First information computing section 220-1 calculates X₁(D) according to

A_(X1,k)(D) of equation 30-1. Similarly, second information computingsection 220-2 calculates X₂(D) according to A_(X2,k)(D) of equation30-2. Similarly, third information computing section 220-3 calculatesX₃(D) according to A_(X3,k)(D) of equation 30-3.

At this time, as described in Embodiment 1, from the conditions tosatisfy in equations 31-1 to 31-g and 32-1 to 32-g, if the coding rateis changed, it is not necessary to change the configuration of firstinformation computing section 220-1, and, similarly, change theconfiguration of second information computing section 220-2 and changethe configuration of third information computing section 220-3.

Therefore, when a plurality of coding rates are supported, by using theconfiguration of the encoder of the highest coding rate as a referenceamong coding rates for sharing encoder circuits, the other coding ratescan be supported by the above operations. That is, regardless of codingrates, LDPC-CCs explained in Embodiment 1 provide an advantage ofsharing first information computing section 220-1, second informationcomputing section 220-2 and third information computing section 220-3,which are main components of the encoder. Also, for example, theLDPC-CCs shown in table 5 provides an advantage of providing data ofgood received quality regardless of coding rates.

FIG. 12 shows the configuration inside first information computingsection 220-1. First information computing section 220-1 in FIG. 12 isprovided with shift registers 221-1 to 221-M, weight multipliers 220-0to 222-M and adder 223.

Shift registers 222-1 to 222-M are registers each storing (where t=0,M), and, at a timing at which the next input comes in, send a storedvalue to the adjacent shift register to the right, and store a valuesent from the adjacent shift register to the left.

Weight multipliers 220-0 to 222-M switch a value of h₁ ^((m)) to 0 or 1in accordance with a control signal outputted from weight controlsection 260.

Adder 223 performs exclusive OR computation of outputs of weightmultipliers 222-0 to 222-M to find and output computation result Y_(1,i)to adder 240 in FIG. 11.

Also, the configurations inside second information computing section220-2 and third information computing section 220-3 are the same asfirst information computing section 220-1, and therefore theirexplanation will be omitted. In the same way as in first informationcomputing section 220-1, second information computing section 220-2finds and outputs calculation result Y_(2,i) to adder 240. In the sameway as in first information computing section 220-1, third informationcomputing section 220-3 finds and outputs calculation result Y_(3,i) toadder 240 in FIG. 11.

Parity computing section 230 in FIG. 11 calculates P(D) according toB_(k)(D) of equations 30-1 to 30-3.

FIG. 13 shows the configuration inside parity computing section 230 inFIG. 11. Parity computing section 230 in FIG. 13 is provided with shiftregisters 231-1 to 231-M, weight multipliers 232-0 to 232-M and adder233.

Shift registers 231-1 to 231-M are registers each storing P_(i−t) (wheret=0, . . . , M), and, at a timing at which the next input comes in, senda stored value to the adjacent shift register to the right, and store avalue sent from the adjacent shift register to the left.

Weight multipliers 232-0 to 232-M switch a value of h₂ ^((m)) to 0 or 1in accordance with a control signal outputted from weight controlsection 260.

Adder 233 performs exclusive OR computation of outputs of weightmultipliers 232-0 to 232-M to find and output computation result Z_(i)to adder 240 in FIG. 11.

Referring back to FIG. 11 again, adder 240 performs exclusive ORcomputation of computation results Y_(2,i), Y_(3,i) and Z_(i) outputtedfrom first information computing section 220-1, second informationcomputing section 220-2, third information computing section 220-3 andparity computing section 230, to find and output parity P_(i) at pointin time i. Adder 240 also outputs parity P_(i) at point in time i toparity computing section 230.

Coding rate setting section 250 sets the coding rate of encoder 200 andoutputs coding rate information to information generating section 210.

Based on a parity check matrix supporting equations 30-1 to 30-3 held inweight control section 260, weight control section 260 outputs the valueof h₁ ^((m)) at point in time i based on the parity check polynomials ofequations 30-1 to 30-3, to first information computing section 220-1,second information computing section 220-2, third information computingsection 220-3 and parity computing section 230. Also, based on theparity check matrix supporting equations 30-1 to 30-3 held in weightcontrol section 260, weight control section 260 outputs the value of h₂^((m)) at that timing to weight multipliers 232-0 to 232-M.

Also, FIG. 14 shows another configuration of an encoder according to thepresent embodiment. In the encoder of FIG. 14, the same components as inthe encoder of FIG. 11 are assigned the same reference numerals. Encoder200 of FIG. 14 differs from encoder 200 of FIG. 11 in that coding ratesetting section 250 outputs coding rate information to first informationcomputing section 220-1, second information computing section 220-2,third information computing section 220-3 and parity computing section230.

In the case where the coding rate is 1/2, second information computingsection 220-2 outputs “0” to adder 240 as computation result Y_(2,i),without computation processing. Also, in the case where the coding rateis 1/2 or 2/3, third information computing section 220-3 outputs “0” toadder 240 as computation result Y_(3,i), without computation processing.

Here, although information generating section 210 of encoder 200 in FIG.11 sets information X_(2,i) and information X_(3,i) at point in time ito “0” according to a coding rate, second information computing section220-2 and third information computing section 220-3 of encoder 200 inFIG. 14 stop computation processing according to a coding rate andoutput 0 as computation results Y_(2,i) and Y_(3,i). Therefore, theresulting computation results in encoder 200 of FIG. 14 are the same asin encoder 200 of FIG. 11.

Thus, in encoder 200 of FIG. 14, second information computing section220-2 and third information computing section 220-3 stops computationprocessing according to a coding rate, so that it is possible to reducecomputation processing, compared to encoder 200 of FIG. 11.

Next, the method of sharing LDPC-CC decoder circuits described inEmbodiment 1 will be explained in detail.

FIG. 15 is a block diagram showing the main components of a decoderaccording to the present embodiment. Here, decoder 300 shown in FIG. 15refers to a decoder that can support coding rates of 1/2, 2/3 and 3/4.Decoder 300 of FIG. 14 is mainly provided with log likelihood ratiosetting section 310 and matrix processing computing section 320.

Log likelihood ratio setting section 310 receives as input a receptionlog likelihood ratio and coding rate calculated in a log likelihoodratio computing section (not shown), and inserts a known log likelihoodratio in the reception log likelihood ratio according to the codingrate.

For example, when the coding rate is 1/2, it means that encoder 200transmits “0” as X_(2,i) and X_(3,i), and, consequently, log likelihoodratio setting section 310 inserts a fixed log likelihood ratio for theknown bit “0” as the log likelihood ratios of X_(2,i) and X_(3,i), andoutputs the inserted log likelihood ratios to matrix processingcomputing section 320. This will be explained below using FIG. 16.

As shown in FIG. 16, when the coding rate is 1/2, log likelihood ratiosetting section 310 receives reception log likelihood ratios LLR_(X1,i)and LLR_(Pi) corresponding to X_(1,i) and P_(i), respectively.Therefore, log likelihood ratio setting section 310 inserts receptionlog likelihood ratios LLR_(X2,i) and LLR_(3,i) corresponding to X_(2,i)and X_(3,i), respectively. In FIG. 16, reception log likelihood ratioscircled by doted lines represent reception log likelihood ratiosLLR_(X2,i) and LLR_(3,i) inserted by log likelihood ratio settingsection 310. Log likelihood ratio setting section 310 insertsfixed-value log likelihood ratios as reception log likelihood ratiosLLR_(X2,i) and LLR_(3,i).

Also, in the case where the coding rate is 2/3, it means that encoder200 transmits “0” as X_(3,1), and, consequently, log likelihood ratiosetting section 310 inserts a fixed log likelihood ratio for the knownbit “0” as the log likelihood ratio of X_(3,i), and outputs the insertedlog likelihood ratio to matrix processing computing section 320. Thiswill be explained using FIG. 17.

As shown in FIG. 17, in the case where the coding rate is 2/3, loglikelihood ratio setting section 310 receives as input reception loglikelihood ratios LLR_(X1), LLR_(X2,i) and LLR_(Pi) corresponding toX_(1,i), X_(2,i) and P_(i), respectively. Therefore, log likelihoodratio setting section 310 inserts reception log likelihood ratioLLR_(3,i) corresponding to X_(3,1). In FIG. 17, reception log likelihoodratios circled by doted lines represent reception log likelihood ratioLLR_(3,i) inserted by log likelihood ratio setting section 310. Loglikelihood ratio setting section 310 inserts fixed-value log likelihoodratios as reception log likelihood ratio LLR_(3,i).

Matrix processing computing section 320 in FIG. 15 is provided withstorage section 321, row processing computing section 322 and columnprocessing computing section 323.

Storage section 321 stores an log likelihood ratio, external valueα_(mn) obtained by row processing and a priori value β_(mn) obtained bycolumn processing.

Row processing computing section 322 holds the row-direction weightpattern of LDPC-CC parity check matrix H of the maximum coding rate of3/4 among coding rates supported by encoder 200. Row processingcomputing section 322 reads a necessary priori value β_(mn) from storagesection 321, according to that row-direction weight pattern, andperforms row processing computation.

In row processing computation, row processing computation section 322decodes a single parity check code using a priori value β_(mn), andfinds external value α_(mn).

Processing of the m-th row will be explained. Here, an LDPC code paritycheck matrix to decode two-dimensional M×N matrix H={H_(nm)} will beused. External value α_(mn) is updated using the following updateequation for all pairs (m,n) satisfying the equation H_(mn)=1.

$\begin{matrix}\lbrack 39\rbrack & \; \\{\alpha_{mn} = {( {\prod\limits_{n^{\prime} \in {{A{(m)}}{\backslash n}}}^{\;}{{sign}( \beta_{{mn}^{\prime}} )}} ){\Phi( {\sum\limits_{n^{\prime} \in {{A{(m)}}{\backslash n}}}^{\;}{\Phi ( {\beta_{{mn}^{\prime}}} )}} )}}} & ( {{Equation}\mspace{14mu} 39} )\end{matrix}$

where Φ(x) is called a Gallager f function, and is defined by thefollowing equation.

$\begin{matrix}\lbrack 40\rbrack & \; \\{{\Phi (x)} = {\ln \frac{{\exp (x)} + 1}{{\exp (x)} - 1}}} & ( {{Equation}\mspace{14mu} 40} )\end{matrix}$

Column processing computing section 323 holds the column-directionweight pattern of LDPC-CC parity check matrix H of the maximum codingrate of 3/4 among coding rates supported by encoder 200. Columnprocessing computing section 323 reads a necessary external value α_(mn)from storage section 321, according to that column-direction weightpattern, and finds a priori value β_(mn).

In column processing computation, column processing computing section323 performs iterative decoding using input log likelihood ratio λ_(n)and external value α_(mn), and finds a priori value β_(mn).

Processing of the m-th column will be explained. β_(mn) is updated usingthe following update equation for all pairs (m,n) satisfying theequation H_(mn)=1. Only when q=1, the calculation is performed withα_(mn)=0.

$\begin{matrix}\lbrack 41\rbrack & \; \\{\beta_{mn} = {\lambda_{n} + {\sum\limits_{m^{\prime} \in {{B{(n)}}/m}}^{\;}\alpha_{m^{\prime}n}}}} & ( {{Equation}\mspace{14mu} 41} )\end{matrix}$

After repeating above row processing and column processing apredetermined number of times, decoder 300 finds an a posteriori loglikelihood ratio.

As described above, with the present embodiment, in a case where thehighest coding rate is (q−1)/q among supported coding rates and wherecoding rate setting section 250 sets the coding rate to (s−1)/s,information generating section 310 sets from information X_(s,i) toinformation X_(q−1,i) as “0.” For example, when supported coding ratesare 1/2, 2/3 and 3/4 (q=4), first information computing section 220-1receives as input information X_(L), at point in time i and calculatesterm X₁(D) of equation 30-1. Also, second information computing section220-2 receives as input information X_(2,i) at point in time i andcalculates term X₂(D) of equation 30-2. Also, third informationcomputing section 220-3 receives as input information X_(3,i) at pointin time i and calculates term X₃(D) of equation 30-3. Also, paritycomputing section 230 receives as input parity P_(i−1) at point in timei−1 and calculates term P(D) of equations 30-1 to 30-3. Also, adder 240finds, as parity P_(i) at point in time i, the exclusive OR of thecomputation results of first information computing section 220-1, secondinformation computing section 220-2 and third information computingsection 220-3 and the computation result of parity computing section230.

With this configuration, upon creating an LDPC-CC supporting differentcoding rates, it is possible to share the configurations of informationcomputing sections according to the above explanation, so that it ispossible to provide an LDPC-CC encoder and decoder that can support aplurality of coding rates in a small computational complexity.

Also, in a case where A_(X1,k)(D) to A_(Xq-1,k)(D) are set to satisfythe above <Condition #1> to <Condition #6> described with the aboveLDPC-CCs of good characteristics, it is possible to provide an encoderand decoder that can support different coding rates in a smallcomputational complexity and provide data of good received quality inthe receiver. Here, as described in Embodiment 1, the method ofgenerating an LDPC-CC is not limited to the above case of LDPC-CCs ofgood characteristics.

Also, by adding log likelihood ratio setting section 310 to the decoderconfiguration based on the maximum coding rate among coding rates forsharing decoder circuits, decoder 300 in FIG. 15 can perform decoding inaccordance with a plurality of coding rates. Also, according to a codingrate, log likelihood ratio setting section 310 sets log likelihoodratios for (q−2) items of information from information X_(r,i) toinformation X_(q−1,i) at point in time i, to predetermined values.

Also, although a case has been described above where the maximum codingrate supported by encoder 200 is 3/4, the supported maximum coding rateis not limited to this, and is equally applicable to a case where acoding rate of (q−1)/q (where q is an integer equal to or greater than5) is supported (here, it naturally follows that it is possible to setthe maximum coding rate to 2/3). In this case, essential requirementsare that encoder 200 employs a configuration including first to (q−1)-thinformation computing sections, and that adder 240 finds, as parityP_(i) at point in time i, the exclusive OR of the computation results offirst to (q−1)-th information computing sections and the computationresult of party computing section 230.

Also, if all of coding rates supported by the transmitting and receivingapparatuses (encoder/decoder) are associated with codes based on themethods described with the present embodiment, by providing anencoder/decoder of the highest coding rate among the supported codingrates, it is possible to easily support coding and decoding in aplurality of coding rates and, at this time, provide an advantage ofreducing computational complexity significantly.

Also, although an example case has been described above where thedecoding scheme is sum-product decoding, the decoding method is notlimited to this, and it is equally possible to implement the presentinvention using decoding methods by a message-passing algorithm such asmin-sum decoding, normalized BP (Belief Propagation) decoding, shuffledBP decoding and offset BP decoding, as shown in Non-Patent Literature 5,Non-Patent Literature 6 and Non-Patent Literature 9.

Next, a case will be explained where the present invention is applied toa communication apparatus that adaptively switches the coding rateaccording to the communication condition. Also, an example case will beexplained where the present invention is applied to a radiocommunication apparatus, the present invention is not limited to this,but is equally applicable to a PLC (Power Line Communication) apparatus,a visible light communication apparatus or an optical communicationapparatus.

FIG. 18 shows the configuration of communication apparatus 400 thatadaptively switches a coding rate. Coding rate determining section 410of communication apparatus 400 in FIG. 18 receives as input a receivedsignal transmitted from a communication apparatus of the communicatingparty (e.g. feedback information transmitted from the communicatingparty), and performs reception processing of the received signal.Further, coding rate determining section 410 acquires information of thecommunication condition with the communication apparatus of thecommunicating party, such as a bit error rate, packet error rate, frameerror rate and reception field intensity (from feedback information, forexample), and determines a coding rate and modulation scheme from theinformation of the communication condition with the communicationapparatus of the communicating party. Further, coding rate determiningsection 410 outputs the determined coding rate and modulation scheme toencoder 200 and modulating section 420 as a control signal.

Using, for example, the transmission format shown in FIG. 19, codingrate determining section 410 includes coding rate information in controlinformation symbols and reports the coding rate used in encoder 200 tothe communication apparatus of the communicating party. Here, as is notshown in FIG. 19, the communicating party includes, for example, knownsignals (such as a preamble, pilot symbol and reference symbol), whichare necessary in demodulation or channel estimation.

In this way, coding rate determining section 410 receives a modulationsignal transmitted from communication apparatus 500 of the communicatingparty, and, by determining the coding rate of a transmitted modulationsignal based on the communication condition, switches the coding rateadaptively. Encoder 200 performs LDPC-CC coding in the above steps,based on the coding rate designated by the control signal. Modulatingsection 420 modulates the encoded sequence using the modulation schemedesignated by the control signal.

FIG. 20 shows a configuration example of a communication apparatus ofthe communicating party that communicates with communication apparatus400. Control information generating section 530 of communicationapparatus 500 in FIG. 20 extracts control information from a controlinformation symbol included in a baseband signal. The controlinformation symbol includes coding rate information. Control informationgenerating section 530 outputs the extracted coding rate information tolog likelihood ratio generating section 520 and decoder 300 as a controlsignal.

Receiving section 510 acquires a baseband signal by applying processingsuch as frequency conversion and quadrature demodulation to a receivedsignal for a modulation signal transmitted from communication apparatus400, and outputs the baseband signal to log likelihood ratio generatingsection 520. Also, using known signals included in the baseband signal,receiving section 510 estimates channel variation in a channel (e.g.ratio channel) between communication apparatus 400 and communicationapparatus 500, and outputs an estimated channel estimation signal to loglikelihood ratio generating section 520.

Also, using known signals included in the baseband signal, receivingsection 510 estimates channel variation in a channel (e.g. ratiochannel) between communication apparatus 400 and communication apparatus500, and generates and outputs feedback information (such as channelvariation itself, which refers to channel state information, forexample) for deciding the channel condition. This feedback informationis transmitted to the communicating party (i.e. communication apparatus400) via a transmitting apparatus (not shown), as part of controlinformation. Log likelihood ratio generating section 520 calculates thelog likelihood ratio of each transmission sequence using the basebandsignal, and outputs the resulting log likelihood ratios to decoder 300.

As described above, according to the coding rate (s−1)/s designated by acontrol signal, decoder 300 sets the log likelihood ratios forinformation from information X_(s,i) to information X_(s-1,i), topredetermined values, and performs BP decoding using the LDPC-CC paritycheck matrix based on the maximum coding rate among coding rates toshare decoder circuits.

In this way, the coding rates of communication apparatus 400 andcommunication apparatus 500 of the communicating party to which thepresent invention is applied, are adaptively changed according to thecommunication condition.

Here, the method of changing the coding rate is not limited to theabove, and communication apparatus 500 of the communicating party caninclude coding rate determining section 410 and designate a desiredcoding rate. Also, communication apparatus 400 can estimate channelvariation from a modulation signal transmitted from communicationapparatus 500 and determine the coding rate. In this case, the abovefeedback information is not necessary.

Embodiment 3

The present embodiment will explain a hybrid ARQ (Automatic RepeatreQuest) in an LDPC-CC code formed by the search method explained inEmbodiment 1.

FIG. 21 shows an example of the frame configuration of a modulationsignal transmitted from communication apparatus #1 (e.g. base stationapparatus) that performs HARQ. In the frame configuration of FIG. 21,the retransmission information symbol refers to a symbol for reportinginformation as to whether retransmission data or new data is provided,to the communicating party (e.g. terminal apparatus). The coding rateinformation symbol refers to a symbol for reporting the cording rate tothe communicating party. The modulation scheme information symbol refersto a symbol for transmitting the modulation scheme to the communicatingparty.

The other control information symbols refer to symbols for reportingcontrol information such as the data length, for example. Also, symbolsfor transmitting information (hereinafter “data symbols”) refer tosymbols for transmitting encoded data (codeword) (e.g. information andparity) acquired by applying LDPC-CC coding to data (information). Thedata symbols include data for detecting frame error such as CRC (CyclicRedundancy Check).

FIG. 22 shows an example of the frame configuration of a modulationsignal transmitted by communication apparatus #2 (e.g. terminalapparatus) that is the communicating party of communication apparatus#1. In the frame configuration of FIG. 22, the retransmission requestsymbol refers to a symbol indicating whether or not there is aretransmission request. Communication apparatus #2 checks whether or noterror occurs in decoded data, and requests a retransmission when thereis error, or does not request a retransmission when there is no error.The retransmission request symbol refers to a symbol for reportingwhether or not there is the retransmission request.

The other control information symbols refer to symbols for transmittingcontrol information such as the modulation scheme, used code, codingrate and data length, to communication apparatus #1 of the communicatingparty, for example. The symbol for transmitting information refers to asymbol for transmitting data (information) to transmit to communicationapparatus #1 of the communicating party.

FIG. 23 shows an example of the flow of frames transmitted betweencommunication apparatus #1 and communication apparatus #2 according tothe present embodiment, referring to HARQ. Also, an example case will beexplained below where communication apparatus #1 and communicationapparatus #2 support coding rates of 1/2, 2/3 and 3/4.

In FIG. 23[1], first, communication apparatus #1 transmits a modulationsignal of frame #1. At that time, data transmitted by the data symbolarea of frame #1 represents a codeword acquired by applying coding of acoding rate of 3/4 to new data.

In FIG. 23[2], communication apparatus #2 receives, demodulates, decodesand performs a CRC check of the modulation signal of frame #1. As aresult of this, there is no error, and therefore a retransmission is notrequested to communication apparatus #1.

In FIG. 23[3], communication #1 transmits a modulation signal of frame#2. Here, data transmitted by the data symbol area of frame #2represents a codeword acquired by applying coding of a coding rate of3/4 to new data.

In FIG. 23[4], communication apparatus #2 receives, demodulates, decodesand performs a CRC check of the modulation signal of frame #2. As aresult of this, there is an error, and therefore a retransmission isrequested to communication apparatus #1.

In FIG. 23[5], communication apparatus #1 receives a retransmissionrequest from communication apparatus #2, and therefore transmits frame#2′ for frame #2. To be more specific, communication apparatus #1encodes part of data (information) using a coding rate of 2/3 lower thana coding rate of 3/4 used upon acquiring a codeword transmitted by frame#2, and transmits only parity of the resulting codeword by frame #2′.

Here, data transmitted by frame #2 and data transmitted by frame #2′will be explained using FIG. 24.

Upon the initial transmission, with frame #2, information X_(1,i),X_(2,i) and X_(3,i) (where i=1, 2, . . . , m) and parity P_(3/4,i)(where i=1, 2, . . . , m) acquired by applying LDPC-CC coding of acoding rate of 3/4 to information X_(2,i) and X_(3,i), are transmitted.

When a retransmission request of frame #2 is requested fromcommunication apparatus #2 to communication apparatus #1, communicationapparatus #1 generates parity P_(2/3,i) (where i=1, 2, . . . , m) byusing a coding rate of 2/3 lower than a coding rate of 3/4 used upon theinitial transmission and encoding X_(1,i) and X_(2,i) (where i=1, 2, . .. , m) among information X_(1,i), X_(2,i) and X_(3,i) (where i=1, 2, . .. , m) transmitted by frame #2.

Also, with frame #2′, only this parity P_(2/3,i)(where i=1, 2, m) istransmitted.

At this time, especially when an encoder of communication apparatus #1is configured in the same way as in Embodiment 2, it is possible toperform both coding of a coding rate of 3/4 upon the initialtransmission and coding of a coding rate of 2/3 upon a retransmission,using the same encoder. That is, even in the case of performing aretransmission by HARQ, it is possible to perform coding upon aretransmission using an encoder to use when performing coding upon theinitial transmission, without adding a new encoder for HARQ.

Thus, upon performing HARQ, if an encoder supports a plurality of codingrates and parity check polynomials for these plurality of coding ratesrepresent the LDPC-CCs described in Embodiment 1, it is possible to usethe same encoder as the one to use when performing coding upon theinitial transmission.

In FIG. 23[6], communication apparatus #2 receives, demodulates, decodesand performs a CRC check of a modulation signal of frame #2′ transmittedupon a retransmission.

The operations in FIG. 23[6] (the method of decoding data upon aretransmission) will be explained using FIG. 25. Upon a retransmission,frame #2′ is decoded using a decoding result of frame #2 receivedearlier.

To be more specific, first, as the initial decoding upon aretransmission (first step), information X_(1,i) and information X_(2,1)(where i=1, 2, . . . , m) are decoded (i.e. LDPC-CC decoding processingof a coding rate of 2/3 is performed), using the LLRs (Log LikelihoodRatios) of information X_(1,i) and X_(2,1) (where i=1, 2, . . . , m)received earlier by frame #2 and the LLR of parity P_(2/3,i) (where i=1,2, . . . , m) of a coding rate of 2/3 received by frame #2′.

In frame #2′, the coding rate is lower than frame #2, so that it ispossible to improve the coding gain, increase a possibility of beingable to decode information X_(1,i) and X_(2,i) (where i=1, 2, . . . ,m), and secure the received quality upon a retransmission. Here, onlyparity data is retransmitted, so that the efficiency of datatransmission is high.

Next, as second decoding upon a retransmission (second step): theestimation values of information X_(1,i) and X_(2,i) (where i=1, 2, . .. , m) are acquired in the first step and therefore are used to generatethe LLRs of information X_(1,i) and X_(2,i) (e.g. an LLR correspondingto “0” of sufficiently high reliability is given when “0” is estimated,or an LLR corresponding to “1” of sufficiently high reliability is givenwhen “1” is estimated); and information X_(3,i) (where i=1, 2, . . . ,m) is acquired by performing LDPC-CC decoding of a coding rate of 3/4using the generated LLRs, the LLR of information X_(3,i) (where i=1, 2,. . . , m) received earlier by frame #2 and the LLR of parity P_(3/4,i)(where i=1, 2, . . . , m) received earlier by frame #2.

In this way, communication apparatus #2 decodes frame #2 transmittedupon the initial transmission, using frame #2′ retransmitted by HARQ. Atthis time, especially when a decoder of communication apparatus #2 isconfigured in the same way as in Embodiment 2, it is possible to performboth coding upon the initial transmission and decoding upon aretransmission (i.e. decoding in the first and second steps), using thesame decoder.

That is, even in the case of performing a retransmission by HARQ, it ispossible to perform decoding upon a retransmission (i.e. decoding in thefirst and second steps) using a decoder used when performing decodingupon the initial transmission, without adding a new decoder for HARQ.

Thus, upon performing HARQ, if an encoder of communication apparatus #1of the communicating party supports a plurality of coding rates andparity check polynomials for these plurality of coding rates representthe LDPC-CCs described in Embodiment 1, it is possible to use the samedecoder as the one to use when performing decoding upon the initialtransmission.

In this way, communication apparatus #2 receives, demodulates, decodesand performs a CRC check of a modulation signal of frame #2′. As aresult of this, there is no error, and therefore a retransmission is notrequested to communication apparatus #2.

In FIG. 23[7], communication apparatus #1 transmits a modulation signalof frame #3. At this time, data transmitted by the data symbol area offrame #3 represents a codeword acquired by applying coding of a codingrate of 3/4 to new data.

In FIG. 23[8], communication apparatus #2 receives, demodulates, decodesand performs a CRC check of the modulation signal of frame #3. As aresult of this, there is an error, and therefore a retransmission is notrequested to communication apparatus #1.

FIG. 26 shows another example of the flow of frames transmitted betweencommunication apparatus #1 and communication apparatus #2 according tothe present embodiment, referring to HARQ. FIG. 26 differs from the flowof frames shown in FIG. 23 in providing a coding rate of 1/2 upon aretransmission and, for frame #2, further retransmitting frame #2″ upona second retransmission in addition to retransmitting frame #2′. Also,an example case will be explained below where communication apparatus #1and communication apparatus #2 support coding rates of 1/2, 2/3 and 3/4.

In FIG. 26[1], first, communication apparatus #1 transmits a modulationsignal of frame #1. At that time, data transmitted by the data symbolarea of frame #1 represents a codeword acquired by applying coding of acoding rate of 3/4 to new data.

In FIG. 26[2], communication apparatus #2 receives, demodulates, decodesand performs a CRC check of the modulation signal of frame #1. As aresult of this, there is no error, and therefore a retransmission is notrequested to communication apparatus #1.

In FIG. 26[3], communication #1 transmits a modulation signal of frame#2. Here, data transmitted by the data symbol area of frame #2represents a codeword acquired by applying coding of a coding rate of3/4 to new data.

In FIG. 26[4], communication apparatus #2 receives, demodulates, decodesand performs a CRC check of the modulation signal of frame #2. As aresult of this, there is an error, and therefore a retransmission isrequested to communication apparatus #1.

In FIG. 26[5], communication apparatus #1 receives a retransmissionrequest from communication apparatus #2, and therefore transmits frame#2′ for frame #2. To be more specific, communication apparatus #1encodes part (or all) of data (information), using a coding rate of 1/2lower than a coding rate of 3/4 used upon acquiring a codewordtransmitted by frame #2, and transmits only parity of the resultingcodeword by frame #2′.

Here, an essential requirement is that: the coding rate used upon aretransmission is lower than a coding rate of 3/4; and, if there are aplurality of coding rates lower than the coding rate used upon theinitial transmission, an optimal coding rate is set among the pluralityof coding rates according to, for example, the channel condition betweencommunication apparatus #1 and communication apparatus #2.

Here, data transmitted by frame #2 and frame #2′ will be explained usingFIG. 27.

Upon the initial transmission, with frame #2, information X_(1,i),X_(2,i) and X_(3,i) (where i=1, 2, . . . , m) and parity P_(3/4,i)(wherei=1, 2, . . . , m) acquired by applying LDPC-CC coding of a coding rateof 3/4 to information X_(1,i), X_(2,i) and X_(3,i), are transmitted.

When a retransmission request of frame #2 is requested fromcommunication apparatus #2 to communication apparatus #1, communicationapparatus #1 generates parity P_(1/2,i) (where i=1, 2, . . . , m) byusing a coding rate of 1/2 lower than a coding rate of 3/4 used upon theinitial transmission and encoding X_(1,i)(where i=1, 2, . . . , m) amonginformation X_(1,i), X_(2,i) and X_(3,i) (where i=1, 2, . . . , m)transmitted by frame #2.

Also, with frame #2′, only this parity P_(1/2,i)(where i=1, 2, . . . ,m) is transmitted.

At this time, especially when an encoder of communication apparatus #1is configured in the same way as in Embodiment 2, it is possible toperform both coding of a coding rate of 3/4 upon the initialtransmission and coding of a coding rate of 1/2 upon a retransmission,using the same encoder. That is, even in the case of performing aretransmission by HARQ, it is possible to perform coding upon aretransmission using an encoder to use when performing coding upon theinitial transmission, without adding a new encoder for HARQ. This isbecause parity check polynomials for a plurality of coding ratessupported by an encoder represent the LDPC-CCs described in Embodiment1.

In FIG. 26[6], communication apparatus #2 receives, demodulates, decodesand performs a CRC check of a modulation signal of frame #2′ transmittedupon a retransmission.

The decoding method upon a retransmission (i.e. upon the firstretransmission) will be explained using FIG. 28. Upon the firstretransmission, communication apparatus #2 decodes frame #2′ using adecoding result of frame #2 received earlier.

To be more specific, first, as the initial decoding upon the firstretransmission (first step), communication apparatus #2 decodesinformation X_(1,1) (where i=1, 2, . . . , m) (i.e. performs LDPC-CCdecoding processing of a coding rate of 1/2), using the LLR ofinformation X_(1,i) (where i=1, 2, . . . , m) received earlier by frame#2 and the LLR of parity P_(1/2,i) (where i=1, 2, . . . , m) of a codingrate of 1/2 received by frame #2′.

In frame #2′, the coding rate is lower than frame #2, so that it ispossible to improve the coding gain, increase a possibility of beingable to decode information X_(1,i) (where i=1, 2, . . . , m), and securethe received quality upon a retransmission. Here, only parity data isretransmitted, so that the efficiency of data transmission is high.

Next, as second decoding upon the first retransmission (second step):communication apparatus #2 acquires the estimation value of informationX_(1,i) (where i=1, 2, . . . , m) in the first step and therefore usesthis estimation value to generate the LLR of information X_(1,i) (e.g.an LLR corresponding to “0” of sufficiently high reliability is givenwhen “0” is estimated, or an LLR corresponding to “1” of sufficientlyhigh reliability is given when “1” is estimated).

Communication apparatus #2 acquires information X_(2,i) andX_(3,i)(where i=1, 2, . . . , m) by performing LDPC-CC decoding of acoding rate of 3/4 using the LLR of information X_(1,i) generated by theestimation value, the LLRs of information X_(2,i) and X_(3,i) (wherei=1, 2, . . . , m) received earlier by frame #2 and the LLR of parityP_(3/4,i) (where i=1, 2, . . . , m) received earlier by frame #2.

In this way, communication apparatus #2 decodes frame #2 transmittedupon the initial transmission, using frame #2′ transmitted by HARQ upona retransmission.

Communication apparatus #2 performs a CRC check of the decoding resultof frame #2. As a result of this, there is error, and therefore aretransmission is requested to communication apparatus #1.

In FIG. 26[7], communication apparatus #1 receives a secondretransmission request from communication apparatus #2, and thereforetransmits frame #2″ for frame #2.

To be more specific, communication apparatus #1 encodes part (or all) ofdata (information) that was not encoded upon the first retransmission,using again a coding rate of 1/2 lower than a coding rate of 3/4 usedupon acquiring a codeword transmitted by frame #2, and transmits onlyparity of the resulting codeword by frame #2″.

Here, data transmitted by frame #2″ will be explained using FIG. 29.

As described above, upon the first retransmission, parity P_(1/2,i)(where i=1, 2, . . . , m) is generated by using a coding rate of 1/2lower than a coding rate of 3/4 upon the initial transmission andapplying LDPC-CC coding of a coding rate of 1/2 to X_(1,i) (where i=1,2, . . . , m) among information X_(1,i), X_(2,i) and X_(3,i) (where i=1,2, . . . , m) transmitted by frame #2 (see FIG. 27). Also, with frame#2′ upon the first retransmission, only this parity P_(1/2,i)(where i=1,2, . . . , m) is transmitted (see FIG. 27).

Upon a second retransmission, parity P_(1/2,i) (where i=1, 2, . . . , m)is generated by using a coding rate (e.g. 1/2) lower than a coding rateof 3/4 upon the initial transmission and applying, for example, LDPC-CCcoding of a coding rate of 1/2 to X_(2,i) (where=1, 2, . . . , m) thatwas not encoded upon the first retransmission among information X_(1,i),X_(2,i) and X_(3,i) (where i=1, 2, . . . , m) transmitted by frame #2(see FIG. 29). Also, with frame #2″ upon a second retransmission, onlythis parity P_(1/2,i) (where i=1, 2, m) is transmitted (see FIG. 29).

Here, LDPC-CC parity check polynomials used upon coding of a coding rateof 1/2 in a second retransmission are the same as LDPC-CC parity checkpolynomials used upon coding of the same coding rate of 1/2 in the firstretransmission (i.e. these are different only in inputs upon coding, butshare the same code used upon coding).

By this means, it is possible to generate a codeword using the sameencoder between the initial transmission and the first retransmission,and further generate a codeword upon a second retransmission using thesame encoder. By this means, it is possible to realize HARQ according tothe present embodiment without adding a new encoder.

In the example shown in FIG. 29, upon a second retransmission, paritycheck polynomials used in coding upon the first retransmission are usedto encode information X_(2,i) (where i=1, 2, . . . , m) other thaninformation X_(1,i) (where i=1, 2, . . . , m) encoded upon the firstretransmission and to transmit the resulting codeword.

In this way, in a case where there are a plurality of retransmissionrequests, upon an n-th retransmission (where n is an integer equal to orgreater than 2), by retransmitting a codeword acquired by preferentiallyencoding information other than information encoded before an (n−1)-thretransmission, it is possible to gradually increase the reliability ofthe log likelihood ratio of each information forming frame #2, so thatit is possible to decode frame #2 more reliably on the decoding side.

Also, in a case where there are a plurality of retransmission requests,upon an n-th retransmission (where n is an integer equal to or greaterthan 2), it is equally possible to retransmit the same data as oneretransmitted before an (n−1)-th retransmission. Also, in a case wherethere are a plurality of retransmission requests, it is equally possibleto further use other ARQ schemes such as chase combining. Also, in acase of performing retransmission several times, the coding rate mayvary every retransmission.

In FIG. 26[8], communication apparatus #2 receives, demodulates, decodesand performs a CRC check of a modulation signal of frame #2″ transmittedagain upon a retransmission (i.e. upon a second retransmission).

The decoding method upon a second retransmission will be explained usingFIG. 30. Upon a second retransmission, communication apparatus #2decodes frame #2″ using a decoding result of frame #2 received earlier.

To be more specific, first, as the initial decoding upon a secondretransmission (first step), communication apparatus #2 decodesinformation X_(2,i) (where i=1, 2, . . . , m) (i.e. performs LDPC-CCdecoding processing of a coding rate of 1/2), using the LLR ofinformation X_(2,i) (where i=1, 2, . . . , m) received earlier by frame#2 and the LLR of parity P_(1/2,i) (where i=1, 2, . . . , m) of a codingrate of 1/2 received by frame #2″.

In frame #2″, the coding rate is lower than frame #2, so that it ispossible to improve the coding gain, increase a possibility of beingable to decode information X_(2,i) (where i=1, 2, . . . , m), and securethe received quality upon a retransmission. Here, only parity data isretransmitted, so that the efficiency of data transmission is high.

Next, as second decoding upon a second retransmission (second step),communication apparatus #2 acquires the estimation value of informationX_(2,i) (where i=1, 2, . . . , m) in the first step, and therefore usesthis estimation value to generate the LLR of information X_(2,i) (e.g.an LLR corresponding to “0” of sufficiently high reliability is givenwhen “0” is estimated, or an LLR corresponding to “1” of sufficientlyhigh reliability is given when “1” is estimated).

Communication apparatus #2 acquires information X_(3,i) (where i=1, 2, .. . , m) by performing LDPC-CC decoding of a coding rate of 3/4 usingthe LLR of information X_(2,i) generated by the estimation value, theLLRs of information X_(3,i) (where i=1, 2, . . . , m) and parityP_(3/4,i)(where i=1, 2, . . . , m) received earlier by frame #2 and theLLR of information X_(1,i) generated by the estimation value ofinformation X_(1,i) (where i=1, 2, . . . , m) estimated in decoding uponthe first retransmission (first and second steps).

In this way, communication apparatus #2 decodes frame #2 transmittedupon the initial transmission, using frame #2′ and frame #2″retransmitted by HARQ.

Communication apparatus #2 decodes frame #2 and then performs a CRCcheck. As a result of this, there is no error, and therefore aretransmission is not requested to communication apparatus #1.

FIG. 31 shows the configuration of communication apparatus #1 thatperforms HARQ according to the present embodiment. Communicationapparatus 600 in FIG. 31 is mounted on, for example, a base station.

Receiving and demodulating section 610 of communication apparatus 600 inFIG. 31 finds a received signal by receiving a modulation signal havingthe frame configuration shown in FIG. 22 transmitted from thecommunicating party, and applies receiving processing such as frequencyconversion, demodulation and decoding to the received signal, therebyextracting a retransmission request symbol. Receiving and demodulatingsection 610 outputs the retransmission request symbol to retransmissionrequest deciding section 620.

Retransmission request deciding section 620 decides from theretransmission request symbol whether or not there is a retransmissionrequest, and outputs the decision result to switching section 640 asretransmission request information. Further, depending on whether or notthere is a retransmission request, retransmission request decidingsection 620 outputs a designation signal to encoding section 650 andbuffer 630.

To be more specific, if there is no retransmission request,retransmission request deciding section 620 outputs a designation signalto encoding section 650 such that encoding section 650 performs codingusing a coding rate set as the coding rate to use upon the initialtransmission. In contrast, if there is a retransmission request,retransmission request deciding section 620 outputs a designation signalto encoding section 650 such that, in the case of selecting HARQ,encoding section 650 performs coding upon a retransmission using acoding rate lower than the coding rate to use upon the initialtransmission (here, in the case of not selecting HARQ, for example, inthe case of selecting chase combining, a coding rate lower than thecoding rate to use upon the initial transmission is not necessarilyselected). Further, if there is a retransmission request, retransmissionrequest deciding section 620 outputs a designation signal to buffer 630such that buffer 630 outputs stored data (information) S20 to switchingsection 640.

Buffer 630 stores data (information) S10 outputted to encoding section650 via switching section 640, and outputs data (information) S20 toswitching section 640 according to a designation signal fromretransmission request deciding section 620.

Switching section 640 outputs one of data (information) S10 and data(information) S20 stored in buffer 630 to encoding section 650,according to retransmission request information. To be more specific, ifthe retransmission request information indicates no retransmissionrequest, switching section 640 outputs data (information) S10 that isnot encoded yet, to encoding section 650 as new data. In contrast, ifthe retransmission request information indicates a retransmissionrequest, switching section 640 outputs data (information) S20 stored inbuffer 630 to encoding section 650 as retransmission data.

Encoding section 650 has encoder 200 shown in Embodiment 2, appliesLDPC-CC coding to input data according to the coding rate designatedfrom retransmission request deciding section 620 and acquires an LDPC-CCcodeword.

For example, in the case of transmitting frame #2 of FIG. 23[3] upon theinitial transmission, encoding section 650 applies coding to informationX_(1,i) X_(2,i) and X_(3,i) (where i=1, 2, . . . , m) using a codingrate of 3/4, according to a designation signal reported fromretransmission request deciding section 620, and generates parityP_(3/4,i) (where i=1, 2, . . . , m) (see FIG. 24).

Then, encoding section 650 outputs information X_(1,i), X_(2,i) andX_(3,i) (where i=1, 2, . . . , m) and parity P_(3/4,i) (where i=1, 2, .. . , m) to modulating and transmitting section 660 as an LDPC-CCcodeword.

Also, for example, in the case of transmitting frame #2′ of FIG. 23[5]upon the first retransmission, encoding section 650 switches the codingrate from 3/4 to 2/3 according to a designation signal reported fromretransmission request deciding section 620, applies coding to X_(1,i)and X_(2,i) (where i=1, 2, . . . , m) among information X_(1,i), X_(2,i)and X_(3,i) (where i=1, 2, . . . , m) transmitted by frame #2, andgenerates parity P_(2/3,i) (where i=1, 2, . . . , m) (see FIG. 24).

Here, an important point is that encoding section 650 includes encoder200 explained in Embodiment 2. That is, in a case where encoder 200performs LDPC-CC coding of a time varying period of g (where g is anatural number) supporting coding rates of (y−1)/y and (z−1)/z (y<z),encoding section 650 generates an LDPC-CC codeword using parity checkpolynomial 42 upon the initial transmission, and, if there is aretransmission request, generates an LDPC-CC codeword using parity checkpolynomial 43 upon a retransmission.

$\begin{matrix}\lbrack 42\rbrack & \; \\{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{z - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 42} )\end{matrix}$

In equation 42, D is a delay operator, and k is an integer.

$\begin{matrix}\lbrack 43\rbrack & \; \\{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 43} )\end{matrix}$

In equation 43, D is a delay operator, and k is an integer. Here,A_(Xf,k)(D) of equation 42 and A_(Xf,k)(D) of equation 43 are equal(where f=1, 2, 3, . . . , y−1), and B_(k)(D) of equation 42 and B_(k)(D)of equation 43 are equal.

By this means, even in the case of performing a retransmission by HARQ,it is possible to perform coding upon a retransmission using an encoderto use when performing coding upon the initial transmission, withoutadding a new encoder for HARQ.

Then, encoder 650 outputs only this parity P_(2/3,i) (where i=1, 2, . .. , m) to modulating and transmitting section 660 as an LDPC-CCcodeword.

Modulating and transmitting section 660 applies transmission processingsuch as modulation and frequency conversion to the LDPC-CC codeword, andtransmits the result to communication apparatus #2 of the communicatingparty via an antenna (not shown).

FIG. 32 shows an example of the main components of communicationapparatus #2, which is the communicating party of communicationapparatus #1. Communication apparatus 700 in FIG. 32 is mounted on, forexample, a terminal apparatus.

Receiving and demodulating section 710 of communication apparatus 700 inFIG. 32 receives as input a received signal via an antenna (not shown),applies radio processing such as frequency conversion to the receivedsignal and acquires a received signal having the frame configurationshown in FIG. 21. Receiving and demodulating section 710 extractscontrol information symbols such as a retransmission information symbol,coding rate information symbol and modulation scheme information symbolfrom the received signal, and outputs these control information symbolsto control information analyzing section 720. Further, receiving anddemodulating section 710 extracts a data symbol from the received signaland outputs the data symbol to log likelihood ratio generating section730 as received data.

From the control information symbols, control information analyzingsection 720 extracts control information of information as to whetherretransmission data or new data is provided, control information of thecoding rate and control information of the modulation scheme, from thecontrol information symbols, and outputs these items of controlinformation to decoding section 740.

Log likelihood ratio generating section 730 calculates the loglikelihood ratio of received data. Log likelihood ratio generatingsection 730 outputs the log likelihood ratio to decoding section 740.

Decoding section 740 has decoder 300 of FIG. 15, decodes the loglikelihood ratio of the received data using control information reportedfrom control information analyzing section 720, and updates the loglikelihood ratio of the received data.

For example, in the case of receiving frame #2 of FIG. 23[3] transmittedupon the initial transmission, decoding section 740 sets the coding rateto 3/4 according to a designation signal reported from controlinformation analyzing section 720 and performs decoding to provide thelog likelihood ratio of received data after decoding processing.

Also, for example, in the case of receiving frame #2′ of FIG. 23[5]transmitted upon a retransmission, decoding section 740 switches thecoding ratio from 3/4 to 2/3 according to a designation signal reportedfrom control information analyzing section 720 and performs decoding toprovide the log likelihood ratio of received data after decodingprocessing. Here, upon a retransmission, decoding section 740 performsdecoding in a plurality of steps. An example case will be explainedbelow where frame #2 of FIG. 23[3] and frame #2′ of FIG. 23[5] arereceived.

To be more specific, first, as the initial decoding upon aretransmission (first step), decoding section 740 decodes informationX_(1,i) and X_(2,i) (where i=1, 2, . . . , m) (i.e. performs LDPC-CCdecoding processing of a coding rate of 2/3), using the LLRs ofinformation X_(1,i) and X_(2,i) (where i=1, 2, . . . , m) receivedearlier by frame #2 and the LLR of parity P_(2/3,i) (where i=1, 2, . . ., m) of a coding rate of 2/3 received by frame #2′.

In frame #2′, the coding rate is lower than frame #2, so that it ispossible to improve the coding gain, increase a possibility of beingable to decode information X_(1,i) and X_(2,i) (where i=1, 2, . . . ,m), and secure the received quality upon a retransmission. Here, onlyparity data is retransmitted, so that the efficiency of datatransmission is high.

Next, as second decoding upon a retransmission (second step), decodingsection 740 acquires the estimation values of information X_(1,i) andX_(2,i) (where i=1, 2, . . . , m) in the first step and therefore usesthese estimation values to generate the LLRs of information X_(1,i) andX_(2,i) (e.g. an LLR corresponding to “0” of sufficiently highreliability is given when “0” is estimated, or an LLR corresponding to“1” of sufficiently high reliability is given when “1” is estimated).

Decoding section 740 acquires information X_(3,i) (where i=1, 2, . . . ,m) by performing LDPC-CC decoding of a coding rate of 3/4 using the LLRsof information X_(1,i) and X_(2,i) generated by the estimation values,the LLR of information X_(3,i) (where i=1, 2, . . . , m) receivedearlier by frame #2 and the LLR of parity P_(3/4,i) (where i=1, 2, . . ., m) received earlier by frame #2.

Here, an important point is that decoding section 740 includes decoder300 explained in Embodiment 2. That is, in a case where decoder 300performs LDPC-CC decoding of a time varying period of g (where g is anatural number) supporting coding rates of (y−1)/y and (z−1)/z (y<z),decoding section 740 decodes an LDPC-CC codeword using parity checkpolynomial 42 in decoding upon the initial transmission, decodes anLDPC-CC codeword using parity check polynomial 43 in decoding upon thefirst retransmission (first step), and decodes an LDPC-CC codeword usingparity check polynomial 42 in decoding upon a second retransmission(second step).

By this means, even in the case of performing a retransmission by HARQ,it is possible to perform decoding upon a retransmission (i.e. decodingin the first and second steps) using a decoder to use when performingdecoding upon the initial transmission, without adding a new decoder forHARQ.

Decoding section 740 outputs the log likelihood ratio of received dataafter decoding processing, to deciding section 750.

Deciding section 750 acquires decoded data by estimating data based onthe log likelihood ratio received as input from decoding section 740.Deciding section 750 outputs the decoded data to retransmissionrequesting section 760.

Retransmission requesting section 760 performs error detection byperforming a CRC check of the decoded data, forms retransmission requestinformation based on whether or not there is error, and outputs theretransmission request information to modulating and transmittingsection 770.

Modulating and transmitting section 770 receives as input data(information) and retransmission request information, acquires amodulation signal by applying processing such as coding, modulation andfrequency conversion to the data and the retransmission requestinformation, and outputs the modulation signal to communicationapparatus #1 of the communicating party via an antenna (not shown).

In this way, with the configurations of FIG. 31 and FIG. 32, it ispossible to implement HARQ of the present embodiment. By this means, itis possible to perform coding upon a retransmission using an encoder touse when performing coding upon the initial transmission, without addinga new encoder for HARQ. Also, it is possible to perform both decodingupon the initial transmission and decoding upon a retransmission (i.e.decoding in the first and second steps), using the same decoder. Thatis, it is possible to perform decoding upon a retransmission (i.e.decoding in the first and second steps) using a decoder to use whenperforming decoding upon the initial transmission, without adding a newdecoder for HARQ.

An aspect of the encoder of the present invention that creates alow-density parity-check convolutional code of a time varying period ofg (where g is a natural number) using a parity check polynomial ofequation 44 of a coding rate of (q−1)/q (where q is an integer equal toor greater than 3), employs a configuration having: a coding ratesetting section that sets a coding rate of (s−1)/s (s≤q); an r-thcomputing section that receives as input information X_(r,i) (where r=1,2, . . . , q−1) at point in time i and outputs a computation result ofA_(Xr,k)(D)X_(i)(D) of equation 44; a parity computing section thatreceives as input parity P¹⁻¹ at point in time i−1 and outputs acomputation result of B_(k)(D)P(D) of equation 44; an adding sectionthat acquires an exclusive OR of computation results of the first to(q−1)-th computation sections and the computation result of the paritycomputing section, as parity P_(i) at point in time i; and aninformation generating section that sets zero between informationX_(s,1) and information X_(q−1,i).

$\begin{matrix}{\mspace{79mu} \lbrack 44\rbrack} & \; \\{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{s - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}} + {\sum\limits_{r = s}^{q - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 44} )\end{matrix}$

In equation 44, D is a delay operator, and k is an integer.

An aspect of the decoder of the present invention that provides a paritycheck matrix based on a parity check polynomial of equation 45 of acoding rate of (q−1)/q (where q is an integer equal to or greater than3) and decodes a low-density parity-check convolutional code of a timevarying period of g (where g is a natural number) using beliefpropagation, employs a configuration having: a log likelihood ratiosetting section that sets log likelihood ratios for information frominformation X_(s,i) to information X_(g−1,i) at point in time i (where iis an integer), to a known value, according to a set coding rate of(s−1)/s (s≤q); and a computation processing section that performs rowprocessing computation and column processing computation according tothe parity check matrix based on the parity check polynomial of equation45, using the log likelihood ratio.

$\begin{matrix}{\mspace{79mu} \lbrack 45\rbrack} & \; \\{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{s - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}} + {\sum\limits_{r = s}^{q - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 45} )\end{matrix}$

In equation 45, D is a delay operator, and k is an integer.

An aspect of the encoding method of the present invention for encoding alow-density parity-check convolutional code of a time varying period ofg (where g is a natural number) supporting coding rates of (y−1)/y and(z−1)/z (y<z), includes: generating a low-density parity-checkconvolutional code of the coding rate of (z−1)/z using a parity checkpolynomial of equation 46; and generating a low-density parity-checkconvolutional code of the coding rate of (y−1)/y using a parity checkpolynomial of equation 47.

$\begin{matrix}\lbrack 46\rbrack & \; \\{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{z - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 46} )\end{matrix}$

In equation 46, D is a delay operator, and k is an integer.

$\begin{matrix}\lbrack 47\rbrack & \; \\{{{{B_{k}(D)}{P(D)}} + {\sum\limits_{r = 1}^{y - 1}{{A_{{Xr},k}(D)}{X_{r}(D)}}}} = {0\mspace{14mu} ( {k = {i\mspace{14mu} {mod}\mspace{14mu} g}} )}} & ( {{Equation}\mspace{14mu} 47} )\end{matrix}$

In equation 47, D is a delay operator, and k is an integer. Here,A_(Xf,k)(D) of equation 46 and A_(Xf,k)(D) of equation 47 are equal(where f=1, 2, 3, . . . , y−1), and B_(k)(D) of equation 46 and B_(k)(D)of equation 47 are equal.

The present invention is not limited to the above-described embodiments,and can be implemented with various changes. For example, although caseshave been mainly described above with embodiments where the presentinvention is implemented with an encoder and decoder, the presentinvention is not limited to this, and is applicable to cases ofimplementation by means of a power line communication apparatus.

It is also possible to implement the encoding method and decoding methodas software. For example, provision may be made for a program thatexecutes the above-described encoding method and communication method tobe stored in ROM (Read Only Memory) beforehand, and for this program tobe run by a CPU (Central Processing Unit).

Provision may also be made for a program that executes theabove-described encoding method and decoding method to be stored in acomputer-readable storage medium, for the program stored in the storagemedium to be recorded in RAM (Random Access Memory) of a computer, andfor the computer to be operated in accordance with that program.

It goes without saying that the present invention is not limited toradio communication, and is also useful in power line communication(PLC), visible light communication, and optical communication.

The disclosures of Japanese Patent Application No. 2008-179636, filed onJul. 9, 2008, and Japanese Patent Application No. 2008-227505, filed onSep. 4, 2008, including the specifications, drawings and abstracts, areincorporated herein by reference in their entireties.

INDUSTRIAL APPLICABILITY

With the encoder, decoder and encoding method according to the presentembodiment, it is possible to realize a plurality of coding rates in asmall circuit scale and provide data of high received quality in anLDPC-CC encoder and decoder.

REFERENCE SIGNS LIST

-   -   100 LDPC-CC encoder    -   110 data computing section    -   120, 230 parity computing section    -   130, 260 weight control section    -   140 mod 2 adder    -   111-1 to 111-M, 121-1 to 121-M, 221-1 to 221-M, 231-1 to 231-M        shift register    -   112-0 to 112-M, 122-0 to 122-M, 222-0 to 222-M, 232-0 to 232-M        weight multiplier    -   200 encoder    -   210 information generating section    -   220-1 first information computing section    -   220-2 second information computing section    -   220-3 third information computing section    -   240 adder    -   250 coding rate setting section    -   300 decoder    -   310 log likelihood ratio setting section    -   320 matrix processing computing section    -   321 storage section    -   322 row processing computing section    -   323 column processing computing section    -   400, 500 communication apparatus    -   410 coding rate determining section    -   420 modulating section    -   510 receiving section    -   520, 730 log likelihood ratio generating section    -   530 control information generating section    -   600, 700 communication apparatus    -   610, 710 receiving and demodulating section    -   620 retransmission request deciding section    -   630 buffer    -   640 switching section    -   650 encoding section    -   660, 770 modulating and transmitting section    -   720 control information analyzing section    -   740 decoding section    -   750 deciding section    -   760 retransmission requesting section

1. A transmission apparatus applied in an optical communication system,comprising: an encoder, which, in operation, outputs a parity sequenceand an information sequence, the parity sequence comprising low-densityparity-check (LDPC) encoded data obtained by LDPC-encoding theinformation sequence using a parity check matrix in which “n” number of(where n is an integer equal to or greater than 1) parity check equationgroups including a plurality of parity check polynomials are arranged,each of the plurality of parity check polynomials satisfying zero; and atransmitter, which, in operation, converts a modulation signal obtainedby modulating the LDPC encoded data into an optical signal and outputsthe optical signal, wherein the parity sequence is generated by using afirst column to a determined column of the parity check matrix for aninformation sequence having a sequence length that corresponds to alength from the first column to the determined column among one or morecolumn(s) of the parity check matrix that corresponds to n-th paritycheck equation group, the determined column varying depending on codingrates, and the “n” number of parity check equation groups support codingrates of (r−1)/r (where r equals to or more than 2 and equals to or lessthan q, where q is a natural number equal to or more than 3), support atime varying period of g (where g is an integer equal to or more 2), andare arranged repeatedly at every g-th row of the parity check matrix,the parity check polynomial being represented as:A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+ . . . +A _(Xq-1,k)(D)X_(q−1)(D)+B _(k)(D)P(D)=0(k=i mod g)  (Equation 1) where, in Equation 1,X₁(D), X₂(D), X_(g−1)(D) are polynomial representations of informationX₁, X₂, . . . , X_(q−1) at time i respectively, P(D) is a polynomialrepresentation of parity P, A_(Xr,k)(D) is a term of X_(r)(D) in theparity check polynomial for “k=i mod g” at time i where the coding rateis (r−1)/r, and B_(k)(D) is a term of P(D) in the parity checkpolynomial for “k=i mod g” at time i, where “i mod g” is a remainderafter dividing i by g.
 2. A transmission method using an opticalcommunication system, comprising: outputting a parity sequence and aninformation sequence, the parity sequence comprising low-densityparity-check (LDPC) encoded data obtained by LDPC-encoding theinformation sequence using a parity check matrix in which “n” number of(where n is an integer equal to or greater than 1) parity check equationgroups including a plurality of parity check polynomials are arranged,each of the plurality of parity check polynomials satisfying zero; andconverting a modulation signal obtained by modulating the LDPC encodeddata into an optical signal and outputting the optical signal, whereinthe parity sequence generated by using a first column to a determinedcolumn of the parity check matrix for an information sequence having asequence length that corresponds to a length from the first column tothe determined column among one or more column(s) of the parity checkmatrix that corresponds to n-th parity check equation group, thedetermined column varying depending on coding rates, and the “n” numberof parity check equation groups support coding rates of (r−1)/r (where requals to or more than 2 and equals to or less than q, where q is anatural number equal to or more than 3), support a time varying periodof g (where g is an integer equal to or more 2), and are arrangedrepeatedly at every g-th row of the parity check matrix, the paritycheck polynomial being represented as:A _(X1,k)(D)X ₁(D)+A _(X2,k)(D)X ₂(D)+ . . . +A _(Xq-1,k)(D)X_(q−1)(D)+B _(k)(D)P(D)=0(k=i mod g)  (Equation 2) where, in Equation 2,X₁(D), X₂(D), X_(g−1)(D) are polynomial representations of informationX₁, X₂, . . . , X_(q−1) at time i respectively, P(D) is a polynomialrepresentation of parity P, A_(Xr,k)(D) is a term of X_(r)(D) in theparity check polynomial for “k=i mod g” at time i where the coding rateis (r−1)/r, and B_(k)(D) is a term of P(D) in the parity checkpolynomial for “k=i mod g” at time i, where “i mod g” is a remainderafter dividing i by g.